The stress sensitivity of shaley sandstones · data have subjected each rock sample to a range of...

Geophysical Prospecting, 2007, 55, 155–168

The stress sensitivity of shaley sandstones

Colin MacBeth1∗ and Christophe Ribeiro2

1Heriot-Watt University, Institute of Petroleum Engineering, Riccarton, Edinburgh, EH14 4AS, UK, and 2CGG, London, SE18NL UK

Received June 2005, revision accepted June 2006

ABSTRACTThe link between the stress sensitivity of shaley sandstones and their porosity andclay content is investigated. This is achieved by firstly fitting a compliance-basedstress-sensitivity law to laboratory measurements of ultrasonic velocity taken fromfour sets of reservoir sandstones, extracted from a variety of depositional settings.Correlations are then sought between the independent parameters of this law andthe porosity or clay fraction of the rocks, which are then subsequently interpreted interms of framework or pore-space-related microstructural clay models. The generalconclusion drawn from the results is that both of the parameters defining the stress-sensitivity law (the asymptotic modulus and the stress-dependent excess compliance)clearly vary with porosity. However, only the asymptotic modulus shows a convincingtrend with clay and there is little observed variation of the stress-dependent compliancewith clay. There is therefore a resultant variation of stress sensitivity with clay, butit is controlled only by the asymptotic modulus. The analysis also concludes that allfour data sets fall into a framework-related category of clay model.

I N T R O D U C T I O N

Despite the numerous variables controlling the elastic prop-erties of a sandstone (such as mineralogy, pore shape andmorphology, grain size, shape and contacts, sorting and pack-ing, degree of consolidation, degree of cementation, confiningpressure, pore fluid, pore pressure and temperature), porosityand clay content remain two of the most popular parameterslinked to the measured elastic moduli of both consolidated andunconsolidated sandstones (see, e.g. Mavko et al. 1998). Thisattention is natural, as the effect of these physical attributesis often of critical importance for reservoir description. Theinfluence of illite cement (subcommercial) versus kaolinite ce-ment (economic) on the permeability of Lower Permian gasreservoirs is an extreme example supporting the choice of claycontent as one of the master variables in reservoir description.In an analogous fashion to permeability, it is believed thata clay fraction of a few percent can also strongly influencethe elastic properties of the rocks (Han et al. 1986). Further-

∗E-mail: [email protected]

more, because clay minerals exist in the rocks as dispersedfine-scale particles or as grain-scale structural and laminarshale deposits, modelling (Minear 1982; Anstey 1991; Samsand Andrea 2001) and measurement (Hornby and Murphy1987) have suggested that the type and location of the claysis also important. Whilst it is still uncertain as to whether ob-servations from laboratory experiments can distinguish rockswith differently located clays, it is generally accepted that anincreasing clay content does tend to decrease the bulk modu-lus, the shear rigidity and the wave velocities by softening therock frame (e.g. Tosaya and Nur 1982; Klimentos 1991). Theinfluence on stress sensitivity is more difficult to understand,and empirical formulations (such as those of Eberhart-Phillips,Han and Zuback 1989) treat the effects of porosity and clayas acting independently of the effective stress. Indeed, theseauthors conclude that no meaningful relationship appears toexist between the effects of the clay, porosity and effectivestress. Here, we take a new look at this result by using an alter-native approach to parameterizing stress sensitivity, achievedby applying the compliance-based law of MacBeth (2004).Our study then uses this law to investigate the relationshipbetween the stress-sensitivity parameters and porosity and/or

C© 2007 European Association of Geoscientists & Engineers 155

156 C. MacBeth and C. Ribeiro

clay by studying four sets of data from previously publishedlaboratory experiments. For the purposes of this current work,shaley sandstone is defined as sandstone that contains suffi-cient clay or silt to affect its measured physical properties. Itshould be noted that this definition differs from that given inthe sedimentology or petrology literature (Selley 1982), but isconsistent with that implicitly assumed in the domains of rockmechanics, rock physics and fluid flow.

D ATA A N A LY S I S

Laboratory database and results

Three shaley sandstone data sets previously published byother authors are chosen for our analysis. These data orig-inate from a range of depositional environments: the upperPalaeocene deepwater channelized turbidite deposits of theWest of Shetlands, UK Continental Shelf (MacBeth 2004) (de-fined as ‘WoS’); Triassic gas-producing sandstones from theCooper Basin, Australia (Khaksar et al. 1999) deposited ineither a fluvio-deltaic setting or meandering point-bar and ae-olian environments (defined as ‘Cooper Basin’); and finallytwo groups of rocks from a mixture of depositional settingsmeasured by Han et al. (1986) (defined as ‘Han GC’ and ‘HanConsolidated’). The original researchers who generated thesedata have subjected each rock sample to a range of petrologi-cal analyses to investigate the clay content. The percentage ofclay in a particular sample can be determined by particle-sizeanalysis using laser granulometry. The mineralogical compo-sition of the rock can be evaluated using X-ray diffraction orFourier-transform infrared spectroscopy (although these donot yield information on the distribution of the individualclays). Finally, authigenic (formed during diagenesis) and de-trital (included during deposition as layers or fragments) claycomponents can be distinguished using thin-section analysiscomplemented by scanning electron microscopy. In principle,the scanning electron microscopy can provide a quantitativeanalysis of the particles, pores, contacts, clay distribution andclay type.

There are 25 samples of the Palaeocene sandstone (WoS)with a generally high porosity of between 23% and 32%.Fourier-transform infrared-spectroscopy mineralogy revealsthat they contain 39% to 71% quartz, 2% to 25% calcite anddolomite, and a range of clays (mainly montmorillonite, illite,kaolinite) up to a clay fraction of 17%. The database of consol-idated Triassic sandstones (Cooper Basin) has low-to-mediumporosities between 3% and 17%. The 22 samples chosen aremainly composed of 60% to 78% quartz, with a clay min-

eral content ranging from 7% to 19% (X-ray diffraction andthin sections) and a small proportion of carbonate. Scanningelectron microscopy identified the clay minerals as mainly au-thigenic kaolinite and illite, but with dominant kaolonite. Thedatabase of Han et al. (1986) has been used elsewhere onseveral occasions (Eberhart-Phillips et al. 1989; Marion et al.1992). Excluding data from rocks with a clay fraction higherthan 25% avoids clay becoming the dominant load-bearingphase and the complications of interpreting a bi-modal modelfor the clay–sand mix (Marion et al. 1992). Note that, as thedata of Han et al. (1986) are obtained from a mixture of depo-sitional environments and conditions, and as scanning electronmicroscopy information is not now available, it is difficult toassess the proportion of detrital and authigenic clays in thesesamples. It is thus decided to subdivide the data into two sub-sets, with the first subset (Han Consolidated) consisting of22 well-consolidated sandstones extracted from quarries andcomposed mainly of kaolinite (determined by point countingfrom thin sections). The second subset (Han GC) of 14 poorlyconsolidated samples from offshore wells in the Gulf of Mex-ico differ in that the clays are mainly composed of illite andmontmorillonite. However, both subsets have a similar rangeof porosity and clay content, and together cover a wide rangeof porosity from 2% to 30%, and have a clay fraction of upto 24%. All the above data sets chosen for our study are roomdry, with the exception of the data of Han et al. (1986), whichare fully water saturated and for which the effect of fluidshave been observed and explained by Han et al. (1986). De-tails of the clay fraction and porosity for each group of rocksare shown in Fig. 1.

The velocity and density data are converted into bulk mod-ulus (κ) and shear rigidity (μ) values as a function of isotropicconfining stress (or pressure P). Elastic moduli are fitted asthese have a more direct relationship to the physics of defor-mation (MacBeth 2004). The theory outlined in the Appendixsuggests a single generic law for separating stress-sensitive andstress-insensitive elastic properties. The following general re-lationships are fitted to the data:

κ (P) = κ∞1 + κ∞ Fκe−P/Pκ

(1)

and

μ (P) = μ∞1 + μ∞ Fμe−P/Pμ

, (2)

where P is the effective pressure, κ∞ and μ∞ are high-pressureasymptotes and Fκ and Fμ are constants directly propor-tional to the total (pressure-dependent) excess compliance ofeach sample at room conditions prior to the experiment (see

C© 2007 European Association of Geoscientists & Engineers, Geophysical Prospecting, 55, 155–168

Stress sensitivity of shaley sandstones, 157

Figure 1 Porosity–clay distributions for the rock data sets analysed in this study. The Han et al. (1986) data have been split into two sets ofconsolidated and unconsolidated sands.

Table 1 Mean asymptotic moduli and excess compliance values determined for thedata used in this study. Due to water saturation, the ratio of bulk to shear modulusis higher in the Han Gulf Coast data set than for the other data sets. Also Fκ is muchlower than Fμ in the Han data sets, due to the stiffening influence of the water

κ∞ Fκ μ∞ Fμ Vclay �

(GPa) (GPa−1) (GPa) (GPa−1) (%) (%) κ∞μ∞

FκFμ

WoS 10.46 0.105 8.31 0.166 6.52 26.98 1.26 0.63Cooper Basin 22.46 0.085 22.39 0.069 11.75 9.68 1.00 1.29Han (Gulf Coast) 20.60 0.019 11.90 0.091 13.36 21.38 1.73 0.21Han (consolidated) 22.62 0.008 16.51 0.029 6.00 17.93 1.37 0.28

Appendix). This model contains the stress dependence for allmechanisms of internal weakness in a single exponential term.The high-pressure asymptotes measure the elasticity of theisotropic background material devoid of any weaknesses. Theprecision of the final curves fitted to the various data (withabsolute errors less than 2%) suggests that, to a good ap-proximation, all rocks can be adequately represented by theseexpressions, regardless of the variety of microstructural mech-anisms. It is our initial expectation that the stress-sensitivityparameters of these equations should now correlate with thevariables controlling the rock properties, such as porosity, clay

type and distribution, and rock mineralogy. Table 1 gives de-tails of the average results for the various data sets considered.

Figs 2–9 show the resultant high-pressure asymptotes M∞and excess compliance-related (F) terms as a function ofporosity and clay fraction. For the WoS data there is a linearporosity dependence of κ∞ and μ∞ with only a small scatter(Fig. 2). When dividing the data into porosity subsets, a slightlinear trend with clay (at fixed porosity) appears for thesemoduli (particularly at low porosity), and there is a clearseparation between lower (18%) and higher (27%) poros-ity groups. Fμ clearly varies with porosity and to a lesser extent



Figure 2 Plots of the high-pressure asymptote parameters, κ∞ and μ∞, as a function of both measured porosity and clay fraction for the WoSsubset of our shaley sandstone database. Points have been colour coded with the second parameter. The data are fitted by interpreted lineartrends for fixed porosity or clay fraction, with the clay dependence separating into a high and a low porosity trend with porosity values asindicated.

Figure 3 Plots of the stress sensitivity parameters, Fκ and Fμ, as a function of both measured porosity and clay fraction for the WoS subset ofour shaley sandstone database. Points have been colour coded with the second parameter. All lines are interpreted trends at fixed porosity orclay fraction. Upper circled points show the suggestion of a separate trend but at a much higher excess compliance.



Figure 4 Plots of the high-pressure asymptote parameters, κ∞ and μ∞, as a function of both measured porosity and clay fraction for the CooperBasin subset of our shaley sandstone database. Points have been colour coded with the second parameter. The data are fitted by interpretedlinear trends at fixed porosity and clay fraction, with the clay dependence showing separate high and low porosity trends.

Figure 5 Plots of the stress sensitivity parameters, Fκ and Fμ, as a function of both measured porosity and clay fraction from the CooperBasin subset of our shaley sandstone database. The data are fitted by interpreted linear trends at fixed porosity and clay fraction, with the claydependence showing separate high and low porosity trends. Circles designate groups not fit fully fitted by the trend lines.



Figure 6 Plots of the high-pressure asymptote parameters, κ∞ and μ∞ ,as a function of both measured porosity and clay fraction from theHan GC subset of the Han shaley sandstone database. Points have been colour coded by the second parameter. The data are fitted by interpretedlinear trends at fixed porosity and clay fraction, with the clay dependence showing separate high and low porosity trends. Points correspond tosimilar rock types sampled in several different wells.

Figure 7 Plots of the stress sensitivity parameters, Fκ and Fμ,as a function of both measured porosity and clay fraction for the Han GC subset ofthe Han shaley sandstone database. The data are fitted by interpreted linear trends at fixed porosity and clay fraction, with the clay dependenceshowing separate high and low porosity trends. The clay trends are debatable, but are drawn to accord with the general behaviour of the otherdata in this study and also the model results. Clay dependence again shows some evidence of high and low porosity subgroups.



Figure 8 Plots of the stress sensitivity parameters, κ∞ and μ∞, as a function of both measured porosity and clay fraction for the Han Consolidatedsand subset of the Han shaley sandstone database. Points have been colour coded with the second parameter. The data are fitted by interpretedlinear trends at fixed porosity and clay fraction, with the clay dependence showing separate high and low porosity trends.

Figure 9 Plots of the stress sensitivity parameters, Fκ and Fμ, as a function of both measured porosity and clay fraction for the Han Consolidatedsubset of the Han shaley sandstone database. The data are fitted by interpreted linear trends at fixed porosity and clay fraction, with the claydependence showing separate high and low porosity trends. A high and a low porosity trend is noticeable in the clay variation for Fμ, althoughthis is less convincing than the other data sets.



with clay (Fig. 3), whilst the Fκ trend is difficult to discern butappears fairly constant. Interestingly, there are some pointswith anomalously high excess compliances. These belong tosamples that do not have unusually different porosity and claycontent; however, one possible cause for these anomalies couldbe varying degrees of core damage.

The κ∞ and μ∞ results for the Cooper Basin data againassume a reasonably linear trend with porosity (Fig. 4). Forthe clay variation, the data split into distinct lower and higherporosity subgroups, with a linearly decreasing trend in each.The subgroups correspond to the different sands from a braid-deltaic setting, meandering point-bar and aeolian environ-ments. The separate porosity subgroups are again apparentfor the excess compliance terms Fκ and Fμ, although there is avirtually constant variation with clay (Fig. 5). However bothF terms increase linearly with porosity.

κ∞ and μ∞ can also be fitted by a straight-line trend inporosity for both the Han Consolidated and the Han GC sand-stones (Figs 6 and 8). However, there is a smaller and lessobvious clay trend than in the previous data sets. As before,for the clay variation the data do still split into two distinctporosity subgroup trends. The effect of water saturation on theasymptotic moduli is not immediately obvious; however, theratio of the averages κ∞/μ∞ for each data set is larger thanfor the Cooper Basin and WoS sandstones (Table 1). This isprincipally due to the stiffening effect of the water on the porespace. Fκ and Fμ increase with porosity in both data sets, butthe data are quite noisy. The clay dependence fits are less sat-isfactory, particularly for the Han GC sandstones, and mightdecrease, increase or remain constant (Figs 7 and 9). The over-all magnitude of the Fκ estimates is significantly smaller thanfor the other data sets, which may again be explained by a gen-eral reduction in the stress sensitivity due to water stiffening.The Fμ estimates are slightly smaller, although less so relativeto Fκ , such that the ratio Fκ/Fμ is still much reduced.

Summing up for the entire selection of data, porosity is, asanticipated, a dominant controlling parameter for the asymp-totic moduli, whilst clay fraction is less influential (see also,e.g. Klimentos 1991). Interestingly, despite the results beingnoisy, the excess compliances are observed to display somedegree of variation with both porosity and clay fraction, suf-ficient on balance to warrant a closer analysis in terms of apossible microstructural model.

Clay microstructural models

We believe that it is not an easy task to classify or distinguishthe behaviour of clays on the overall elastic properties, particu-

larly the stress sensitivity, without a complete understanding ofthe rock’s origin in terms of deposition, and subsequent com-paction and diagenesis. For example, syndepositional illite willdiffer in its mechanical properties from illite originating fromdiagenetic alteration of kaolinite. However, even with full pet-rographic details available, it may still be difficult to be definiteabout the effect of the clays as the action of the geological pro-cesses is so varied and complex: there are many possible geo-logical scenarios and depositional settings that can lead to thesame class of mechanical behaviour as observed at the labora-tory scale. As a consequence, we choose to define a generic mi-cromechanical classification, and examine its consequence onmeasurements observed at the laboratory scale. Anstey (1991)provides a good description of the underlying reasoning forthis approach, and Bryant and Raikes (1995) take a similarroute but use contact theory and void space aspect ratio in-stead of compliances. In this model, we distinguish betweenthe stress-independent components of the rock which remainunaltered during loading in the laboratory, and the collectiveof stress-sensitive components which make up the weaknessesinherent in the rock (such as loose grain-grain contacts, in-tragranular cracks, small aspect-ratio pore space, weakenedrock fragments, and possibly clays). All of these weaknesseshave one common feature, i.e. with increasing pressure/stressthey effectively diminish to zero (assuming no internal dam-age during deformation). Thus, this model is defined by aneffective high-pressure asymptote M∞ (representing either κ∞or μ∞), and an overall excess pressure-dependent complianceF (representing either Fκ or Fμ) in terms of the most compli-ant elements associated with the rock frame or the effectivepore space (the interconnected pore volume that contributesto fluid flow). Pressure dependence of the porosity due to com-paction should also be included in the pore-space complianceterms. The two stress-independent contributions to the elasticcompliance are:� Zpore

0 – from the pore structure;� Z clay0 – from the clays.

In addition, there are three stress-sensitive excess compli-ances� Z frame (P) – the ‘lithology’ contribution from the rock-frame

material, arising from random rock fragments and possiblecracks or thin isolated pores. This is related to the diageneticand tectonic history of the rock and is fixed by such factorsas the shape and angularity of the grains, packing and sort-ing, grain transport mechanism, energy of the depositionalsystem, and degree of pressure solution. Due to its nature,this factor is defined as being independent of the porosityand clay content.



� ZporeP (P) – representing that part of the pore space between

loose contacts or enclosed low aspect-ratio pore space. Theexcess compliance of these features is directly proportionalto porosity or any other physical features related to porosity.� Z clay

P (P) – a possible direct link to the clay contribution,which we separate out for the purposes of the analysis.All of the compliance quantities are defined at room condi-

tions and per unit volume except Z frame, which is a fixed quan-tity for a given rock. They are expected to have different rela-tive magnitudes for different rock types and state when testedin the laboratory. Based on the above, the following classesdefine the elastic moduli and dominant stress-sensitivity char-acter of the rocks:

Class Ia – framework-related: embedded shale grains aredeposited as framework fragments amongst the quartz grains,maintaining the general pore-space size and geometry. In thiscategory we include lithic fragments and interstitial clays. Asthese grains also support the load (albeit with an elasticallysofter modulus), the rocks have a mechanical behaviour simi-lar to that of clean sandstones. The resultant stress-sensitivityproperties are defined generically by

F = Z frame + �

1 − �Z pore

P , (3)

where the porosity � is the effective (connected) porosityas measured at room conditions by a helium or mercuryporosimeter. The dependence on porosity is defined such thatthe compliance is infinite when � → 1. In this particularmodel, clays do not significantly change the pore-space com-pliance as there is no overall re-shaping of the internal geome-try; therefore, the asymptotic moduli are explicitly dependentonly on porosity. Following Nur et al. (1995) and the resultsof MacBeth (2004),

M∞ = Mm(1 − �Z pore

0 Mm). (4)

The factor �Zpore0 is the total stress-independent compli-

ance associated with the pore space. The frame modulus Mm

is a composite of the quartz, clay minerals, cements and otherminerals. Mm may be calculated by mixing the mineralogy andcements using an appropriate law such as the Voigt-Reuss-Hill average or embedding the clay with Kuster–Toksoz the-ory applied as by Minear (1982). As the clay elastic mod-uli are much smaller than those of the quartz-cement matrix(Vanorio et al. 2003), the sandstone matrix Mm will be soft-ened by the addition of clay, and therefore decreases slightlywith clay fraction (Vclay) (Tosaya and Nur 1982; Klimentos1991).

Class Ib – as in class Ia, except that the shale grains (and alsolayers or lenses) are loosely bonded and so create an additionalstress dependence. This may occur because the shale particlesdo not bond or couple well to the quartz or because theirinternal structure is more stress sensitive than the quartz. Thisleads to additional clay-related stress sensitivity, Z clay

P per unitframe volume, that modifies only the excess compliance term:

F = Z frame + �

1 − �Z pore

P + Vclay Z clayP , (5)

where Vclay is the volume fraction of clay in the rock. Theasymptotic moduli remain unaltered from equation (4). Forlaminar clay deposits, an alternative explanation for the stress-sensitivity contribution Z clay

P could be the anisotropic defor-mation of these clays due to the internal microporosity. If thisis the case, then these small aspect-ratio pores may also con-tribute an additional compliance component in the asymptoticmoduli M∞ :

M∞ = Mm

(1 −

(�Z pore

0 + Vclay Z clay0

)Mm

). (6)

Class II – pore-space-related: this class relates to authigenicclays that line the grain surfaces (chlorite, smectite or illite),growing across the pore space from the grain surfaces (illite)or generally filling pores and cementing grain contacts (kaoli-nite). Such clays, when introduced into the pore space after theframework has been deposited may slightly affect the porosityand hence the density, but not the elastic moduli or their stresssensitivity. However, if these clays cement the grains togetherand line pores, they may provide an overall strengthening ofthe pore space (Anstey 1991), increasing its stiffness and re-ducing its stress sensitivity by an amount proportional to Vclay.This modifies the modulus term,

M∞ = Mm

(1 −

(�Z pore

0 − Vclay Z clay0

)Mm

), (7)

and also the excess compliance term,

F = Z frame + �

1 − �Z pore

P − Vclay Z clayP , (8)

where Z clayP is now the cumulative reduction in compliance.

To illustrate the predictions from these models, variationsfor porosity (10–25%) and clay fraction (0–15%) are shownin Figs 10(a,b) for the asymptotic moduli and F-values, re-spectively. The curves were generated using compliance valuesthat will later be seen to be typical of the data sets under con-sideration. For M∞, we use Zpore

0 = 0.08 GPa−1 and Z clay0 =

0.02 GPa−1, whilst for the F-values, we use ZporeP =

0.30 GPa−1, Z clayP = 0.20 GPa−1 and Z frame = 0.05 GPa−1. For

the modulus data, Mm is calculated by assuming a pure quartzframe (κ = 37 GPa) and use is made of the Voigt-Reuss-Hill



Figure 10 (a) Expected variation in porosity for fixed clay content and vice versa, for the asymptotic moduli M∞. Parameters are chosen tobest typify the data considered in this study as defined by the excess compliance values Z pore

0 and Z clay0 of Table 2. Trends which best represent

the individual data sets are marked. (b) Expected variation in porosity for fixed clay content and vice versa, for the stress sensitivity F-values.Parameters are chosen to best typify the data considered in this study as defined by the excess compliance values, Z frame, Z pore

P and Z clayP of

Table 2. Trends which best represent the individual data sets are marked.

average to determine the consequence of mixing the quartzwith the clay. For the clay moduli, upper limit values obtainedfrom measurements of montmorillonite and kaolinite are used(κ = 12 GPa) (Vanorio et al. 2003). It is predicted that for allclasses, M∞ should decrease linearly with porosity, and thatthe clay content has little influence on this variation. This isin general agreement with the observations in Figs 2,4,6 and8 where the trends for the porosity variation are fairly tightlyclustered around a single linearly decreasing trend. The vari-ation with clay content is less pronounced; however, there isthe possibility of a change of slope: compare, for example,the curves for low and high porosity in class Ia, or the pre-dictions of class Ib versus those of class II. As expected, theobserved clay variation does show a clear separation into dif-ferent trends for each porosity subgroup with little overallchange in slope. By contrast, F is predicted to increase linearlywith porosity and exhibit moderate incremental changes in

intercept due to clay fraction. Again, this is a trend observedacross all of our data sets (see Figs 3,5,6 and 7), althoughthe intercept changes due to clay are not immediately appar-ent. For the plots of F against clay fraction, class Ia rocks arepredicted to show no variation whilst classes Ib and II possessopposite signs of slope and could therefore perhaps provide ev-idence with which to distinguish unequivocally between thesetwo clay models. However, as the slopes of F against Vclay arein general small, there are no clear examples of this in ourdata. One feature of particular interest is the possible changein the sign of the gradient between Fκ (class Ia) and Fμ (classIb) in the West of Shetland data (Fig. 3), although the Fκ valueis not well constrained. In the next section, the observationswill be analysed in more detail and the individual compliancecomponents estimated. In the data, there are small groups ofoutliers to the marked trends; it is thought their origin couldbe a distinctly different Zframe value.



Figure 10 Continued.

Model fits to data

The selected laboratory data are now analysed further in thelight of the models proposed in the previous section. In all fourcases, the models of equations 3 to 8 are fitted (where appro-priate) to the porosity and clay dependence, and the individualcompliance values in the M∞ and F terms are determined fromthis procedure. Visually fitted straight-line trends are drawnthrough the set of points, guided by templates of variationssimilar to those shown in Figs 10(a,b), and defined by spe-cific compliance parameters. These trends are determined firstin the �-domain for a fixed mean clay fraction, and then inthe Vclay-domain for several fixed mean porosity values, withthe aim of making simultaneous fits to both the porosity andclay variations using a common set of compliance values. Thisprocedure proved, on the whole, to be sufficient to yield rea-sonably unique and reliable sets of individual compliance pa-rameters (Table 2). In this analysis, the clay and pure quartzframe moduli assigned are those used in the predictions of theprevious section, i.e. κ = 12 GPa and μ = 6 GPa for clay andκ = 37 GPa and μ = 44 GPa for the quartz. As above, the

Voigt-Reuss-Hill average is also used to determine the resultof mixing the quartz with the clay.

Comparison of Figs 10(a,b) with the laboratory results ofFigs 2 and 3, respectively, for the WoS shaley sandstones indi-cates that these data broadly follow the predictions of class Ib.Although a negative Z clay

P value for the stress sensitivity factorFκ is suggestive of class II, the Zpore

P value for Fμ dominates allother contributions; this perhaps suggests tangential deforma-tion across loose grain-grain contacts between the quartz orclay particles as a predominant source of the observed stresssensitivity. The asymptotic moduli, κ∞ and μ∞, appear almostequally affected by both the elastic compliance of the clay Z clay

0

and pore space Zpore0 (see Table 2). Thin section or scanning

electron microscopy images are not available to us for theserocks; however, their geological age and unconsolidated stateis consistent with the interpretation that the clay minerals formpart of a framework-related shale component and this impliesa class I model.

For the Cooper Basin sandstones, the compliance param-eters (Figs 4 and 5) give clear porosity and clay variationswith less scatter than the previous data set. These are mostly



Table 2 Excess compliance values estimated from fitting the modelpredictions from equations 3 to 8 to the four data sets considered inour analysis. (a) Compliances for the bulk modulus; (b) compliances forthe shear modulus. The compliance estimates in red are more poorlyconstrained due to scatter in the data. Those with a negative sign forZ clay

P are indicative of class II rather than class I(a)

κ clay Z pore0 Z clay

0 Z frame Z poreP Z clay

P(GPa) (GPa−1) (GPa−1) (GPa−1) (GPa−1) (GPa−1)

WoS 12.00 0.070 0.030 0.070 0.020 −0.010Cooper Basin 12.00 0.090 0.002 0.000 0.550 0.000Han GC 12.00 0.060 0.001 0.012 0.050 −0.060Han Consol 12.00 0.060 0.001 0.000 0.040 0.000

(b)

μclay Z pore0 Z clay

0 Z frame Z poreP Z clay

P(GPa) (GPa−1) (GPa−1) (GPa−1) (GPa−1) (GPa−1)

WoS 6.00 0.070 0.040 0.000 0.350 0.080Cooper Basin 6.00 0.090 0.002 0.000 0.450 0.000Han GC 6.00 0.080 0.040 0.000 0.350 0.000Han Consol 6.00 0.080 0.010 0.000 0.140 0.000

in agreement with class Ia, the assignment being based onthe virtually constant values of Fκ and Fμ with clay. Thestress sensitivity once more appears to be controlled mainlyby pore-space-related features through Zpore

P , again suggest-ing that the origin is loose framework supporting grainslinked to the porosity. Although petrological analyses based onthin sections and scanning electron microscopy (Khaksar andGriffiths 2000) give the dominant clay type in the CooperBasin samples as dispersed authigenic kaolonite with microp-orosity, lithic rock fragments are also detected as a strong sec-ondary component in their composition (with their fractionlinked to the overall sample porosity); these may therefore bethe origin of this stress sensitivity.

The results for the Han GC sandstones (Figs 6 and 7) sug-gest class Ia, but perhaps hint at class II, categorized princi-pally through Fκ , although the scatter of the data points makesdefinite assignment difficult. The stress sensitivity shows aprominent Zpore

P value for the shear deformation Fμ. The elas-tic compliances are close in value to those of the WoS sand-stones, indicating a similar physical state of the rocks. Finally,the Han Consolidated sandstones show an overall behaviourin agreement with class Ia or Ib (Figs 8 and 9). Confidencein this classification is slightly greater than the Han GC dataset, due to the lower data scatter. The elastic compliance esti-mates are similar to those for the Han GC data set, althoughthe clay-related component of the asymptotic modulus Z clay

0

is lower. Apart from the obvious lower absolute magnitudeof the elastic compliances for the Han saturated data in gen-eral, there are no unusual features in the estimates that sug-gest a difference in the nature of the clays or matrix due tointeraction with the water.

C O N C L U S I O N S A N D D I S C U S S I O N

It is generally concluded that the shaley sandstone data studiedin this work show a dependence of their stress sensitivity onporosity and clay content. In particular, the stress-sensitivitycomponents defined using the compliance parameters, Fκ andFμ, appear to be largely dominated by the pore-space contribu-tion. A porosity-related dependence might be partly explainedby compaction of the samples under the applied load. How-ever, given that the magnitude of this effect for most competentsandstones is small (less than 2%) (MacBeth 2004), it is un-likely that this will account for the strong variation observedhere. Therefore, a more plausible explanation is the cumula-tive effect of loose grain-grain contacts and compliant porespace. Clay content is observed to have little or no effect onFκ and Fμ at the noise levels of the current data sets, althoughthere is an identifiable porosity correlation. In contrast, theasymptotic moduli, κ∞ and μ∞, display a dependence onboth porosity and clay content. Thus, whilst clay does not di-rectly affect the stress-dependent compliance terms, it does still



induce a change in the overall stress sensitivity by virtue of itsalteration of the asymptotic moduli (see equations 1 and 2).

In principle, it appears possible to distinguish between detri-tal (framework) and authigenic (pore-space-related) clay mod-els by interpreting the joint variation of Fκ and Fμ with clayfraction and porosity. However, despite this prediction, it hasnot been possible to demonstrate this using the current datasets, as all rocks appear to be either class Ia or Ib, thus signify-ing that the origin of their properties is detrital. It is also pos-sible that noise in the data may have led to a failure to identifythe separate models. Furthermore, it has not been possible tovalidate these findings using supporting information acquiredfrom first-hand experience, due to the nature of the availabledata set. To identify the different models with more precisionrequires laboratory data with a higher level of detail than hasbeen made available for our current work. We require accessto rock samples that are carefully screened on a sedimentolog-ical basis, and with the petrology of the rock precisely definedand all external factors affecting the rock properties reducedor controlled.

A C K N O W L E D G E M E N T S

This work was funded by the Edinburgh Time-Lapse ProjectPhase II and is submitted with approval from its sponsors: BP,Shell Expro, BG Group, Chevron-Texaco, Conoco-Phillips,Exxon-Mobil, Statoil, Total, Maersk, Norsk-Hydro and Land-mark. CM is grateful to Zhiyu Xu, Bob Keys, Jean-MarieQuestiaux, Andy Gardiner, Andrew Parker and FernandoMoraes for stimulating discussions on the petrological aspectsof this work.

R E F E R E N C E S

Anstey N.A. 1991. Velocity in thin section. First Break 9, 449–457.Bryant S. and Raikes S. 1995. Prediction of elastic-wave velocities in

sandstones using structural models. Geophysics 60, 437–446.Eberhart-Phillips D., Han D.-H. and Zoback M.D. 1989. Empirical

relationships among seismic velocity, effective pressure, porosity,and clay content in sandstone. Geophysics 54, 82–89.

Han D., Nur A. and Morgan D. 1986. Effects of porosity and claycontent on wave velocities in sandstones. Geophysics 51, 2093–2107.

Hornby B.E. and Murphy W.F. 1987. VP/VS in unconsolidated oilsands: Shear from Stoneley. Geophysics 52, 502–513.

Khaksar A. and Griffiths C. 2000. Effects of porosity and clay con-tent on P- and S-wave velocities in Cooper Basin sandstones. Ex-ploration Geophysics 31, 433–440.

Khaksar A., Griffiths C.M. and McCann C. 1999. Compressionaland shear-wave velocities as a function of confining stress in drysandstones. Geophysical Prospecting 47, 487–508.

Klimentos T. 1991. The effects of porosity-permeability-clay contenton the velocity of compressional waves. Geophysics 56, 1930–1939.

MacBeth C. 2004. A classification for the pressure sensitivity proper-ties of a sandstone rockframe. Geophysics 69, 497–510.

Marion D., Nur A., Yin H. and Han D. 1992. Compressional velocityand porosity in sand-clay mixtures. Geophysics 57, 554–563.

Mavko G., Mukerji T. and Dvorkin J. 1998. The Rock Physics Hand-book. Cambridge University Press, ISBN. 0521620686.

Minear J. 1982. Clay models and acoustic velocities. 57th AnnualTechnical Conference, AIME, SPE, SE 11031.

Nur A., Mavko G., Dvorkin J. and Gal D. 1995. Critical porosity: thekey to relating physical properties to porosity in rocks. 65th SEGMeeting, Houston, USA, Expanded Abstracts, 878–881.

Sams M. and Andrea M. 2001. The effect of clay distribution on theelastic properties of sandstones. Geophysical Prospecting 49, 128–150.

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A P P E N D I X

Compliance/compressibility-based model

MacBeth (2004) formulates the pressure dependence of therock frame by introducing excess compliance into an other-wise rigid rock frame. Following Sayers and Kachanov (1995),the rock volume is assumed to be composed of a competentbackground matrix (porous quartz rock frame) populated byinternal weaknesses (these could represent, for example, thepore-space compressibility, loose grain-grain contacts, cracks,rock fragments or perhaps clays). Mathematically, the rockis treated as a homogeneous elastic medium specified by theelastic stiffness tensor cijkl or more appropriately by its inverse,the elastic compliance tensor Sijkl. Internal weaknesses intro-duce a pressure-dependent excess compliance �Sijkl (P) intothe pressure-independent background compliances S∞

ijkl. Theeffective compliance tensor that links macroscopic stress tomacroscopic strain is

Si jkl = S∞i jkl + �Si jkl (P) . (A1)

For an isotropic medium, the bulk and shear moduli κ andμ can be written (MacBeth 2004)

κ (P) = 13 (S1111 (P) + 2S1122 (P))

(A2)



and

μ (P) = 12 (S1111 (P) − S1122 (P))

. (A3)

For the excess compliance due to concentrations of ran-domly distributed and randomly orientated crack or contact-like weaknesses, κ and μ can then be re-written exactly as

κ (P) = κ∞1 + 3κ∞ (�S1111 (P) + 2�S1122 (P))

(A4)

and

μ (P) = μ∞1 + 2μ∞ (�S1111 (P) − �S1122 (P))

, (A5)

where

�S1111 = 3ZN (P) + 2ZT (P)5

(A6)

and

�S1122 = 3 (ZN (P) − 2ZT (P))15

. (A7)

The parameters κ∞ and μ∞ are the background high pres-sure asymptotes, and ZN and ZT are the sum of all normaland tangential compliances in the distribution of weaknesses,weighted according to their individual areas. These compli-ance quantities are functions of pressure, and each must di-minish as pressure increases before finally tending to zero atlarge pressures. In this model, there is a natural reference point

at the high-pressure compressed state for which all internalweaknesses are assumed to be completely absent. Substitutingequation (A6) and (A7) into equation (A4) and (A5) yields

κ (P) = κ∞1 + κ∞ Zκ (P)

(A8)

and

μ (P) = μ∞1 + μ∞ Zμ (P)

, (A9)

where Zκ = 3ZN and Zμ = (6ZT + 4ZN)/5. This model hasbeen successfully tested and fitted to pressure-dependent mod-uli from a large database of laboratory measurements (Mac-Beth 2004) using an exponential model for the compliancedecay.

If more than one type of weakness is present, the compli-ance functions in equation (A6) and (A7) can be broken downinto their constituent parts. Thus, for example, if pore com-pressibility, cracks, loose contacts and clays are each partlyresponsible for the stress sensitivity, then

Z (P) = Z pore + Z crack + Z contacts + Z clay. (A10)

In the main text, contributions to the stress sensitivity arecombined into composite terms related to the frame compo-nent Z frame and the pore-space terms Zpore and Z clay. On theassumption of an exponential decay for each individual con-tributing factor in (A10), their summation is also assumed tobe a rough approximation to an exponential.


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