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Geophysical Prospecting, 2011, 59, 90–110 doi: 10.1111/j.1365-2478.2010.00897.x

Exploring the effect of meso-scale shale beds on a reservoir’s overallstress sensitivity to seismic waves

Colin MacBeth∗, Yesser HajNasser, Karl Stephen, and Andy GardinerInstitute of Petroleum Engineering, Heriot-Watt University, Riccarton Park, Edinburgh EH14 4AS, UK

Received August 2009, revision accepted April 2010

ABSTRACTIntra-reservoir, sub-seismic shale beds with a thickness range from 1–10 m are presentin most clastic reservoirs. They are often studied to investigate their effect on fluidflow and reservoir performance. Here, it is found that shales with such thicknessescan also strongly affect the effective elastic wave behaviour of the composite reservoir,particularly when observed with frequent time-lapse surveys. More specifically, forreservoirs experiencing pressure depletion, the effective impedance of the reservoirinterval is usually expected to harden, however our results indicate that shales canreduce this hardening effect or perhaps unexpectedly soften the overall impedance.The overall in situ stress sensitivity of the reservoir is reduced below that predictedfrom taking laboratory core plug measurements of sand stress sensitivity alone. Thesepredictions are based on a combination of geomechanical and pressure diffusionphenomena that are in turn controlled by the shale thickness, permeability and themechanical properties. As sub-seismic shale layers of approximately 1 m thicknesstake less than three months to pressure equilibrate whilst thicker shale layers of 10 mcan take over 20 years, in the context of repeated seismic surveying our predictionsrequire accurate knowledge of the shale properties and statistics and hence a gooddescription of the sedimentology. Based on our default property values, it appearsthat reduced or anomalous stress sensitivity is likely to be more important for 4Dprojects with frequent acquisitions of 3–12 months but is of less concern when seismicis repeated on conventional time periods of every 5–10 years. The critical set ofparameters required to carry out accurate calibration of these predictions is not yetfully available from published literature.

Key words: 4D seismic, Shales, Stress sensitivity.

ABBREVIATION S USE D

α – generic effective stress coefficientαsa – effective stress coefficient for sandαsh – effective stress coefficient for shaleBsh – Skempton coefficient for shale

ct – total compressibility of the rock and fluidsδij – Kronecker delta

∗E-mail: [email protected]

D – diffusivity coefficient�P – pressure change

�Psa – pressure change in sand�Psh – pressure change in shale�σ tot – total internal stress change (both vertical and horizon-tal)�σ tot

V – total vertical confining stress�σ

effV – vertical effective stress loading rock frameε – strainφ – fractional porosity

γ sa – vertical stress arching ratio for sand

90 C© 2010 European Association of Geoscientists & Engineers

Exploring the effect of meso-scale shale beds 91

γ sh – vertical stress arching ratio for shaleγ v – generic vertical stress arching coefficientγ h – generic horizontal stress arching coefficientP – pressure variable

P0 – initial fluid pressurePsh – fluid pressure in the shalePsai – initial fluid pressure in the sandPsaf – final fluid pressure in the sand

kr – relative permeabilityk – absolute permeabilityL – thickness of the shaleλ – fluid mobility

NTG – net-to-gross in the reservoir intervalMav – average of P-wave moduli for sands and shalesMsa – P-wave elastic modulus for sandMsh – P-wave elastic modulus for shale

μ – viscosityν – Poisson’s ratio

ρav – average of densities for sands and shalesρsa – bulk density of sandρsh – bulk density of shale

R – ratio of channel thickness to widthτ – time constantt – time variable

U – displacementρ0, �ρ , ρρ – parameters for density stress sensitivity lawκ∞, Pκ , Eκ – parameters for stress sensitivity of bulk modulus

μ∞, Pμ, Eμ – parameters for stress sensitivity of shear modu-lus

INTRODUCTION

A precise quantitative prediction of the in situ variationin the elastic properties of the reservoir’s rock-mass withproduction-induced changes is an essential element of thepetroelastic model that links saturation and pressure changesto the corresponding 4D seismic signatures. Currently, how-ever, the practice of using laboratory measurements to cali-brate this variation appears adequate only at a qualitative leveland suitable for 4D seismic feasibility studies or crude visualinterpretation of the 4D seismic signatures. In particular, theexact magnitude of the in situ stress sensitivity of the reservoiris still largely uncertain and estimates cannot be relied uponfor accurate determination of the pressure changes (Eiken andTondel 2005). This point has been recently highlighted by theresults of Floricich et al. (2006) and Stephen and MacBeth(2006), who combined seismic observations with engineeringdata to conclude that the reservoir’s stress sensitivity observed

at the seismic scale is smaller than that anticipated from thelaboratory. A mismatch between predictions and observationsis also reported in other 4D seismic data sets such as those ofFletcher (2004), who revealed an unexplained difference in thesign of the amplitudes and time-shifts associated with deple-tion in gas condensate reservoirs. A possible explanation forthe inability to exactly replicate the 4D seismic observations isoffered by a number of well-documented factors that may actto reduce or enhance the stress sensitivity (MacBeth 2004).The (extensive but not exhaustive) list of factors includes: in-accuracies in specifying the ‘dry’ rock frame properties dueto sample conditions and preparation in the laboratory; inter-nal damage due to cutting and stress unloading if controlledstresses and pore pressure are not applied; the statistical biasintroduced by the core plug sampling scheme due to coreheterogeneity; exact specification of the effective stress coeffi-cient; the true triaxial stress state of the reservoir; creep; andfrequency dispersion effects.

An additional item for consideration in the above list is theexistence of intra-reservoir scale heterogeneities. In clastic sys-tems an important heterogeneity is the distribution of shale. Inparticular, this study focuses on discrete intra-reservoir shalesthat exist as shale beds smaller than the size of a typical sandbody (20–30 m) and much smaller than the seismic wave-length (50–100 m). For the purposes of this paper we refer tosuch shale beds as ‘sub-seismic’. It is known that the seismicresponse cannot, in general, detect these beds but instead re-sponds as if the reservoir is a single homogeneous unit. Theend objective of this work is therefore to examine the effect ofthese shales on this overall, composite sand-shale seismic re-sponse. As this study focuses on reservoir changes monitoredover time periods of several months to tens of years by 4Dseismic surveys, the many shale beds of thicknesses less thanapproximately one metre are not considered – this choice willbe observed below to relate to the effect of pressure diffu-sion. A natural upper limit for investigation is those shalesthicker than 10 m, which can be detected in the seismic andfor which the effective property principle does not apply. Inthe literature, sub-seismic shales are defined to be ‘determin-istic’ or ‘stochastic’ (Fig. 1). Whilst deterministic shales arecontinuous and predictable between wells, uncertainty arisesfor stochastic shales as it is difficult to locate and identify theiroccurrence and length precisely within the inter-well volumefrom well data alone. A number of past studies have consid-ered the impact of stochastic shales or shale breaks on thereservoir’s fluid flow behaviour, well performance and resid-ual oil calculation (for example, Haldorsen and Lake 1984).Sub-seismic shales have also been observed to strongly affect

C© 2010 European Association of Geoscientists & Engineers, Geophysical Prospecting, 59, 90–110

92 C. MacBeth et al.

Figure 1 In this work thin sub-seismic shales are considered, simi-lar in character to that shown above. a) Example of intercalcatingshales from the turbiditic Frigg field (Skaug and Gundesco 1986). b)Thin sub-seismic shales can be distinguished as laterally continuous‘deterministic’ shales or the more laterally discontinuous ‘stochastic’shales.

reservoir performance and as a consequence the 4D seismic re-sponse (for example, McLellan et al. 2006). In these exampleshowever, shales are treated as inactive and inert barriers per-turbing flow in the reservoir and their influence on pressureis largely neglected. Treating shales as barriers is a conve-nient simplification from the fluid flow perspective but maynot necessarily be a valid model of reality. In geomechanicalstudies, sub-seismic shales have also been studied for their im-pact on well failure (Schutjens et al. 2005), although they areconsidered to be non-compacting mainly because their proper-ties cannot be adequately calibrated. Building on this previouswork, our current study focuses on the prediction of the reser-voir’s in situ elastic wave stress sensitivity and the evolution ofthe fluid pressure component. For this purpose, it is necessaryto consider the shales as fully active geomechanical, dynamicand elastically stress sensitive elements of the reservoir. Weconcentrate only on the effects of pressure depletion as a con-sequence of an imbalance in pressure support, using a seriesof idealized geological models.

PREDICTION OF THE EFFECTOF PRESSURE D EPLETION ON S HALES

To understand how the sub-seismic shales affect the reservoirstress sensitivity and hence the 4D seismic response, the me-chanical and dynamic behaviour of the shales are consideredseparately at first and then combined.

The prediction for impermeable but mechanically activeshales

Here, it is assumed that the shale beds are completely imper-meable and therefore act as non-conducting barriers to fluidflow in the reservoir. This assumption is commonly made (andjustified) for most reservoir simulation studies. Consider theeffect of instantaneous pressure depletion in a sand body con-sisting of several sand and shale beds sandwiched between ashale overburden and underburden as in Fig. 2(a). All of thesands are assumed to be connected to the well and hence arefully produced. Good horizontal and reasonable vertical con-nectivity within the sands ensures that the resulting pressuredrop spreads quickly and the pressure reaches a stable statewithin a few seconds. Although in favourable circumstancesthe majority of the fluid volume removed is replaced by theaquifer or injectors, we consider there to be an imbalance andhence a net pressure drop. The individual sand beds are ex-pected to compact in response to this depletion because theeffective stress acting on their rock frame has increased. Intra-reservoir shales, however, cannot respond to the depletionin the same way as the sands, as fluid is not extracted fromthem (at least not over the time scale of production). Theydo nevertheless interact indirectly and because the shales staycoupled to the sands there will be deformations and total stresschanges in the shales also. Thus, in a similar way to the morewidely recognized shale overburden deformation in responseto reservoir compaction (for example, Hodgson et al. 2007),intra-reservoir or inter-reservoir shales may mechanically ex-tend in response to pressure depletion in the sand. Figure 2(b)from the published numerical modelling example of Sayersand Schutjens (2007) illustrates this effect.

To treat the mechanical effect on the shale beds, considerthe influence of a pressure depletion of �P on the elastic waveproperties of the reservoir (�P is defined as negative for adrop in pressure). For each sand bed, the reduction in reser-voir fluid pressure leads instantaneously to an overall stresschange of αsa�P relative to the pre-production state (whereαsa is the effective stress coefficient describing the contributionof the fluid pressure to the overall stress – it is defined as a



Figure 2 a) Schematic illustrating the impact of production on a sequence of sands separated by shales. b) Modelled vertical strain alternationat the location of a well in a deepwater Gulf of Mexico turbidite as a function of true vertical depth (TVD) (Sayers and Schutjens 2007).

positive number). Upon sand compaction in response to thisstress change, the total vertical stress state inside the reservoiris changed by γ sa�P (where γ sa is the stress arching ratio – seeAppendix A for further details, γ values are defined as positivenumbers). This changes the effective stress acting on the sand-stone rock frame by an amount (γ sa – αsa)�P, where γ sa < αsa.Impermeable shales, on the other hand, are not subjected topressure change due to fluid production and hence their stressstate changes by virtue of the pull exerted by the neighbour-ing compacting sand layers – this leading to a change in totalstress acting internally on the shale. The total stress change inthe shale due to this mechanism (also equal to its change ineffective stress) is given by γ sh�P, where γ sh is the stress arch-ing ratio for the shales. In addition to the above, a secondaryeffect occurs in the shales – as they extend, the pore volumeexpands and the pore pressure decreases by an amount de-termined by their Skempton Bsh coefficient (Detournay andCheng 1993, equation (10)). The change in effective stress onthe shale rock frame is given by (γ sh − (αsh Bsh/3))�P, how-ever as B coefficients are typically similar in magnitude to theeffective stress coefficients, the combination αsh Bsh/3 is cho-sen to be neglected in the discussion here and also below in thenext two sections as it is small compared to γ sh. The seismic-scale consequence of an extension-induced stress change inthe shales is illustrated in the schematic of Fig. 3. The stresssensitivities of the sand and shale rock frame may be defined

by the characteristic non-linear function observed and mea-sured in the laboratory for single core plugs (MacBeth 2004).The sands and shales are assumed to have a similar stresssensitivity behaviour (although a different magnitude) whencompacting but different elastic moduli (red curve segment),but the shale has an enhanced stress sensitivity upon exten-sion (blue curve segment) as the magnitude upon elongationis known to exceed that of compaction (Sayers 2007). Upondepletion, the sands follow the red curve and increase theirimpedance, whilst the shales follow the solid blue curve anddecrease their impedance. Due to the non-linear shape of thestress sensitivity curves (which in practice requires adequatecalibration), for any given pressure drop the shales may re-duce their impedance by an amount greater than the increaseof impedance in the sands due to compaction. Thus the impactof the shale stress sensitivity could be a strong component ofthe reservoir’s overall stress sensitivity.

To quantify the impact of the above phenomena on theoverall seismic response of the reservoir, the change in theelastic wave properties of the sand body are calculated forvertical P-wave propagation through the central portion ofthe body containing a stack of sand and shale beds. For this,Backus (1962) and a mass balance calculation are used to de-termine the pre-production composite P-wave elastic modulusMav and density ρav from the individual moduli Msa and Msh,and densities ρsa and ρsh, for the sands and shales initially



Shale extension

Sand compaction

Shale

compaction

Figure 3 Schematic stress sensitivity curves for reservoir sand and shale, illustrating the effect of pressure depletion and compaction versusdilation or extension on the elastic properties. Dashed segments indicate portions of the curves not thought to be accessed by the rocks in situ.γ sh and γ sa are the stress arching factors and αsa is the effective stress coefficient for the sand.

when reservoir pressure is P0

1Mav(bef ore)

= NTGMsa(P0)

+ 1 − NTGMsh(P0)

, (1)

ρav(bef ore) = NTGρsa(P0) + (1 − NTG)ρsh(P0), (2)

and also the modulus and density for the post-production state

1Mav(a f ter )

= NTGMsa(P0 + |(γsa − αsa)�P|)+ 1 − NTG

Msh(P0 − |γsh�P|) ,(3)

and

ρav(a f ter ) = NTGρsa(P0 − |(γsa − αsa)�P|)+ (1 − NTG)ρsh(|P0 − γsh�P|). (4)

Here, NTG refers to the net-to-gross (ratio of cumulative sandthickness divided by the total thickness of the sand body) forthe sand body at a particular horizontal location and the γ sa

and γ sh are the depth averaged quantities at that location. Asonly vertical velocity for P-waves is considered, these resultsare sensitive only to changes in the vertical effective stress.P-wave impedance is calculated from equations (1)–(4) for atypical, good quality (25 per cent porosity) sand from a nor-mally pressured reservoir in the North Sea with top reservoirlying at 7000 ft (similar to that of the Nelson and Schiehallion

fields in the UKCS). An instantaneous pressure depletion of�P = −10 MPa is chosen, this representing a reasonable up-per limit on the pressure fluctuation in an oil-water reservoirdue to production with partial support. Note that in prac-tice larger local variations are still possible depending on thedegree of compartmentalization, type of reservoir and ratesof production. Sand stress sensitivity parameters and elasticproperties published for the Schiehallion field by MacBeth(2004) are used. As measurements of shale stress sensitivityare not available for this area, this property is selected arbi-trarily by manual adjustment of the shale parameters to befirstly greater than that of the sand, then identical to that ofthe sand and finally to be stress insensitive (see Table 1 fordetails). Note that from published literature it appears thatthe shale stress sensitivity upon compression and extensionmay be smaller than that of sands (for example, Jones andWang 1981; Sarout and Guegen 2008), if this is the case thenour results for only the latter two conditions may bracketreality. Density variation with stress is included for complete-ness, despite the magnitude being no more than 0.1–0.2%of the initial value. The final numerical results are shown inFig. 4 for the three shale stress sensitivity scenarios and as afunction of the vertically defined net-to-gross. To select valuesfor γ sa and γ sh in our specific case, extensive geomechanicalmodelling is carried out for a range of sand channel model



Table 1 Elastic wave stress-sensitivity parameters for the sands and shales used in this work, defined according to the parameters of MacBeth(2004). κ∞ and μ∞ are high pressure asymptotes of the behaviour, Eκ and Eμ control the magnitude of the effect and Pκ and Pμ control theonset of the low pressure non-linear behaviour. The pressure relationship is given by ρ(P) = ρ0 + �ρe−P/Pρ

κ∞ Pκ Eκ μ∞ Pμ Eμ ρ0 Pρ �ρ

(GPa) (MPa) No units (GPa) (MPa) No units (g/cm3) (MPa) (g/cm3) Comments

Sand 9.94 6.32 1.00 7.75 7.23 1.22 1.91 8.40 −0.15

Shale 1 8.94 10.32 1.00 6.25 10.23 1.22 2.10 8.40 −0.15 Identical stresssensitivity as thesands

Shale 2 9.94 6.32 1.00 7.75 7.23 1.22 1.91 8.40 −0.15 Less stresssensitivity than thesands

Shale 3 9.94 − 0.00 7.75 − 0.00 1.91 − 0.00 More stresssensitivity than thesands

dimensions, geometries, mechanical properties and shale beddistributions (see Appendix A for details). These results indi-cate that γ sa ≈ γ sh and also that the γ ’s lie mainly between0.05 (shale thickness of 10 m) and 0.5 (shale thickness of 1 m)for our particular model selection. For this reason, results forthe lower and upper extremes of the γ ’s are used as input intothe depth averaged quantities used to generate Fig. 4. Whilstit is understood that the effective stress coefficient αsa is likelystress dependent and less than one for most consolidated sands(for example, Christensen and Wang 1985), to avoid specify-ing a particular rock in our current work it is assumed to beunity.

The results shown in Fig. 4 indicate that there may only be acomplete guarantee of an increase in the reservoir impedancewith pressure depletion when the shales are stress insensitive.Indeed, when the shales have identical stress sensitivity to thesands, or are more stress sensitive, the intuitive interpretationof reservoir ‘hardening’ (i.e., increase in the overall P-waveimpedance) is valid only above a certain threshold net-to-gross value (in this case it is 0.8 for 1 m thick shales and 0.20for 10 m thick shales). The greater the amount of shale in thecomposite reservoir package, the greater this ‘softening’ (i.e.,decrease in overall P-wave impedance) effect with pressuredepletion. Clearly, however, a limit will be reached where thegeomechanical effects of the sands on the inter-bedded shalesis insufficient to cause the necessary extension to induce thiseffect. Softening results in a dimming of the top reservoirresponse instead of a brightening (or vice-versa), a result con-trary to our prior conceptual understanding. Further, for anygiven net-to-gross, the shale extension effect reduces the stresssensitivity of the reservoir below that anticipated for a homo-

geneous and connected sand body. Note that a similar resultis possible if only a small proportion of the sand beds are hy-draulically connected or they become more shaley or possessa high clay content. When there are only a very few thin sandsin the reservoir it is unlikely that their combined strain wouldbe sufficient to produce the desired change in the shales (for afixed pressure drop in the sands). The current theory does notanticipate this lower net-to-gross limit. Thus it is anticipatedthat for very small values of net-to-gross the above calculationmight be unrepresentative of the true physics of deformation.

In summary, considering the geomechanical effects aloneand the assumptions of linearity, elasticity, isotropy and ho-mogeneity inherent in this analysis, it appears that whetherthe reservoir hardens or softens with depletion depends onthe net-to-gross, together with the exact stress sensitivity, me-chanical properties, stress arching ratios γ sa and γ sh and theeffective stress coefficient αsa.

The prediction for permeable but mechanically inactiveshales

In fluid flow simulations, the assumption of impermeableshales is usually based on consideration of the fluid flow rates.As fluids usually leak through shales over only thousand tomillions of years i.e., geological time scales (unless fractured),intact shales are effectively recognized as reservoir seals andbarriers for production time scales (years to tens of years).Shales do have fairly moderate porosities (5–20%) but theyare known to possess very small permeabilities and thus highcapillary entry pressures that inhibit fluid mobility. Typicalpermeabilities reported in the literature are in the range 1 μD



Figure 4 Calculated percentage changes in P-wave impedance (�I) for a reservoir comprising of a mixture of sand and impermeable shale layersundergoing mechanical changes due to a pressure depletion of 10 MPa. For the generation of these figures γ sa and γ sh are equal and are fixedat two values: 0.05 (corresponding to a shale thickness of 10 m in Appendix A) and 0.5 (shale thickness of 1 m). The effective stress coefficientαsa is set to unity in both cases. These results correspond to a reservoir pressure for a normally pressured reservoir at 7000 ft.

to 1 nD, this being many orders of magnitude below that forthe coarser grained clastics that make up the hydrocarbonreservoir. These values are thought to arise from the smallgrain size and hence pore throat size (5–60 nm), tortuous porestructure, unimodal pore size distribution and particular min-eralogical packing/sorting/fabric of the shales (see, for exam-ple, Neuzil 1994). Figure 5 gives a general qualitative measureof permeability for sands, silts, claystone and their interme-diate rocks drawn from a range of relevant available litera-ture. Unfortunately, permeability values for reservoir shalesare problematic due to the lack of reliable core data and mea-

surement techniques but also the associated cost in obtainingaccurate and representative laboratory measurements. In fact,since several orders of magnitude in range are possible fora single porosity value in apparently similar samples, eventhe exact controlling mechanisms for shale permeability arenot yet fully understood (Yang and Aplin 2007). Such diver-sity may arise due to permeability anisotropy linked to par-ticle alignment and material heterogeneity, fractures, internalstructure and laminations of the sand, silt or clay componentsand shale maturity. Hence, for the purposes of this currentstudy the broadest spectrum of permeability values, 1 μD to



Figure 5 General range of permeabilities reported in the literature for shales and associated rocks. Measurements are for reservoir rocks over arange of depths, diverse geographical locations and for both laminated and massive shales. Values are extracted from Howard (1991), Katsube(2000), Best and Katsube (1995) and Schloemer and Krooss (1997). ‘Shales’ are defined as having more than 50% of terrigeneous clasticcomponents <0.0625 mm and thus bracket a range of possible rocks. A more precise classification of shales is unclear as none have wideacceptance.

1 nD, must be considered and the implications in terms of thestress sensitivity of the shale properties addressed accordingly.

The shale permeability is low but finite and thus can af-fect the pressure evolution but not necessarily large-scale fluidmovement. To investigate this more fully, consider a homoge-neous sand-filled reservoir in which is embedded a single thinshale bed. Instantaneous depletion is induced in the sands dueto production and the pressure change is quickly establishedin the (high permeability) sands over only a few seconds. Afterthis, a vertical gradient is then set up across the top and baseof the shale, which in turn drives the pressure equilibrationprocess that occurs in the shale due to its finite permeabil-ity (see Appendix B or Fig. B.1). Although, being relativelyimpermeable compared to the surrounding sand, the shale isstill potentially capable of reaching pressure equilibrium withthe reservoir sands. A key question is the exact time scale forthis process, which depends on both the thickness of the shaleand its absolute permeability. Calculations based on the 1Dpressure diffusivity equation are given in Appendix B for areservoir consisting of a single shale layer embedded betweentwo sands, these yielding the results of Fig. 6. These show thata 1 m thick 1 nD shale takes months to equilibrate, whereasa 10 m thick 1 nD shale can take over tens of years. Thesetimes scale inversely with permeability and thus will be 1000times smaller for 1 μD shales. Thus, in a reservoir it appearsthat 1 nD permeability shales thinner than a metre will befully equilibrated in the time between the baseline and mon-itor survey for most 4D projects (5–10 years repeat time). If,

however, the shales are closer to 10 m thick then there maybe partial equilibration and the results of the previous sectionon impermeable shales become relevant.

The above results (which do not yet include the effects ofgeomechanical extension of the shales) indicate that shales willdeplete partially or totally according to a pressure diffusionphenomenon. The prediction in this case is therefore that allshales will ultimately compact with depletion in the sands andhence the whole reservoir interval will eventually harden – thedegree and speed at which this occurs depends on the shalethickness and diffusivity properties.

The prediction for permeable and geomechanically activeshales

It has been concluded from the previous section that intra-reservoir, sub-seismic shales with a small but finite perme-ability can partially deplete over time scales typical of mostrepeated seismic surveys. The anticipated degree of shale de-pletion however depends on the shale thickness and proper-ties and the specific time scale involved. Depending on theconditions, shales may become completely depleted, partiallydepleted or remain close to their initial pressure. Depletion off-sets the geomechanical extension effects driven by the pressure(and hence stress) difference across the sand-shale boundary.Total stress change in response to sand depletion is (almost)instantaneous, whereas the pressure diffusion acts over a muchlonger time period. It is anticipated that the shales will firstly



Figure 6 Relaxation time for a thin shale sandwiched between two instantaneously depleted sands. Numbers in the box refer to the shalethickness in centimetres. The relaxation time is defined as the point at which the pressure perturbation induced in the shale by the process ofdiffusion is 9/10th of the perturbed value in the sand (t = 2.3τ ). This is derived from the solution of the 1D pressure diffusivity equation (seeAppendix B).

experience an extension in response to the depleted sands andthis will be followed by a slow re-compaction with a timeconstant of months to tens of years. If the shales can toleratethis mechanical behaviour without failure, then the net conse-quence of the process depends, once again, on the time scaleinvolved and the shale thickness. Thinner shales equilibratequickly but also experience a greater geomechanical responseinitially. Thicker shales equilibrate at a slower rate but ex-perience a smaller initial mechanical impulse. The extent ofinteraction between the geomechanical and pressure diffusionphenomena is quantified below. A smaller effect may be notedagain, for completeness, as arising from the Skempton coef-ficient described above. Mechanical extension of the shalescauses a small reduction in the pore pressure, which in turnslightly reduces the rate of equilibration. This effect is notconsidered in the current analysis.

To analyse the combined effects of geomechanics and pres-sure diffusion, numerical calculation of the deformation ofsand bodies with different geometries and aspect ratios is per-formed for a variety of intra-reservoir shale thicknesses anddistributions. This deformation analysis is performed inde-pendently of the pressure diffusion calculation. A separatecalculation is performed for each shale pressure equilibrationstate, ranging from �Psh = 0 (no shale depletion) to �Psh =�P (shales fully depleted with pressure drop being that inthe sand). Note that when the pressure in the shales starts toequilibrate (�Psh �= 0), it becomes necessary to also specify the

effective stress coefficient, αsh, for the shale – thus the changeof effective stress for the shale is now (γ sh�P − αsh�Psh)rather than γ sh�P in the shales. The geomechanical schemeused is that of Geertsma (1973) implemented according toHu (1989) and is identical to the nucleus of strain summationfor a half-space described in Appendix A. The stress archingratios for the sand γ sa and shales γ sh are evaluated for eachdepletion state during the continuous diffusion process andthus can also be mapped as curves defined as a function oftime. The results (see Fig. 7a,b) indicate that the stress arch-ing ratios for the shales vary with time (and hence depletion)and the net-to-gross and those for the sand follow a roughlysimilar trend (although the variation with the net-to-gross isslight and thus not shown). The non-linear nature of the de-crease in γ sh and γ sa is due to the particular shale distributionand thicknesses in the sand body. The numerical calculationsshow that as pressure equilibrates the γ sh values eventuallydecrease below αsh

�Psh�P . The point at which this occurs once

again depends on the shale thickness and can vary betweenthree months for the 1 m thick shale to 20 years for a 10 mthick shale. There is a smaller, secondary variation of thispoint with net-to-gross. The cross-over point varies with theparticular values assigned for the stress arching ratios and ef-fective stress coefficients. In these particular calculations αsh

is set at 0.7 – the effective stress coefficient of shale is poorlydocumented and this is obtained from the low porosity shaleresults of Hornby (1996).



Figure 7 Variation of the stress arching ratios γ sh and γ sa with time for: a) a shale thickness of 1 m and permeability of 1 nD; b) a shalethickness of 10 m and permeability of 1 nD. Note that the time to reach any given depletion state varies depending on the shale thickness andpermeability. The effective stress coefficient αsh for the shale is selected as 0.7 whilst for the sand αsa is 1.0. The calculations used to generatethese curves are made continuously for every pressure change. For visual clarity only, γ sa points for a fixed net-to-gross of 0.7 are drawn insteadof curves.

Figure 8 shows the implications of variable γ sh (and γ sa)on the overall elastic properties of the reservoir. Whilst thesands continue to compact as before, the shales will expe-rience extension at first and the composite elastic responsewill be in accordance with Fig. 3. However as time progressesγ sh reduces below αsh

�Psh�P and the shales reverse their geome-

chanical behaviour and start to compact as the effective stressapplied to the shales increases. In practice this process is a con-tinuous accommodation of the geomechanical effects by theprogression of pressure diffusion as described in Fig. 9, suchthat the strain field in the shale slowly reverses its behaviourand evolves towards its final state. It is a fully coupled processand needs a fully coupled geomechanical and flow simulatorto verify these findings, which is beyond the scope of the cur-rent work. Once again, the exact pathway of this progressiondepends on the shale properties – thickness and permeability.To understand what final impedance may be found by com-

bining geomechanical and diffusion effects, a calculation isperformed again using the Backus averaging equations (1)–(4)for one of our idealized geological models. Stress arching ra-tios are computed numerically using the same scheme as thatdescribed in Appendix A for a sand channel of dimensions25 m × 250 m. Shales with a thickness 1 m and permeabilityof 1 nD are embedded in the sand channel and their numbervaried to simulate a range of net-to-gross. After depletion inthe sands, the time taken for these particular shale layers toequilibrate to within 10 per cent of the final sand pressure is82 days. The results of impedance calculation for two differentstates of pressure equilibration and elapsed time are shown inFig. 10(a,b). The two pressure states considered are: �Psh =0.1 �Psa and �Psh = 0.3 �Psa, these occurring after two daysand seven days respectively and the γsh ≤ αsh

�Psh�P condition is

achieved by the time 47–53 days have elapsed (when �Psh =0.7 �P). Thus, beyond approximately 50 days the reservoir



Figure 8 Stress sensitivity curves for reservoir sand and shale as in Fig. 3 but at a later time after pressure equilibration has occurred and γ sh isnow less than αsh

�Psh�P .

Figure 9 Strain paths for the shales and sand in an instantaneously depleted sand body. The shales slowly evolve to a strain condition similarto the compaction in the sands, crossing over between an extensional and compactional regime.



Figure 10 Time-lapsed change in impedance for a reservoir composed of sands and shales distributed with a varying net-to-gross. As in Fig. 4,the stress sensitivity of the shale properties is chosen to be twice as great as the sand, the same and then stress insensitive. Results are for shales ofthickness 1 m and permeability 1 nD. Results are after an elapsed time of: a) two days and 10% depletion and b) seven days and 30% depletion.

zone is expected only to harden. By contrast, 10 m thick shalestake 250 days to equilibrate to �Psh = 0.1 �Psa, 520 days toreach �Psh = 0.3 �Psa and approximately 4500 days to reachthe �Psh = 0.7 �Psa point. It can be concluded that, for a timeperiod of months or greater a reservoir with 1 m thick shaleswill not exhibit an anomalous softening response, whilst forthe 10 m thick shales there is a possibility of overall reservoirsoftening. However, the likelihood of observing this softeningeffect diminishes over time. For example, five years after thedepletion change it is predicted to be visible only for the 10 mthick shales and in regions with net-to-gross values of 0.2.As a general principle, unless the shales in the reservoir arefully equilibrated then it is expected that the effects describedabove will lead to a reduction in the overall stress sensitiv-

ity from that expected for a fully connected, homogeneousbody.

DISCUSS ION OF THE G EOLOGICALIMPLICATIONS OF THIS STUDY

It is concluded from the previous section that sub-seismicshales may be important when evaluating the overall dynamicstress sensitivity of the reservoir during 4D analysis. However,to determine when and where the effects predicted in the pre-vious sections actually impact requires an understanding ofthe fluid flow properties and thickness of the shales, which inturn originate from the sediment depositional system for eachreservoir. In this work we treat the reservoir as a moderately



thick sandstone-dominated interval containing laterally con-tinuous or discontinuous shales one to ten metres thick. Suchsand bodies are known to occur in a range of depositional en-vironments, including channelized turbidites, onlapping tur-bidite sheets and multistorey/multilateral fluvial sand bodies.Unfortunately, published numerical data specifically on shalethickness are generally scarce, although statistics from out-crops or general summaries for shale length and their varia-tion are readily available (Weber 1982). Indeed, most studiestend to focus on the lateral continuity of shale and the impactof shale breaks or clay/silt intercalations on the fluid transportproperties, especially the vertical flow. Unfortunately, the lat-eral dimensions of the intra-reservoir shale are not a reliableway of evaluating its thickness. This point is highlighted byHaldorsen and Lake (1984) in a modelling-based study basedon data from a Cambrian reservoir in Algeria. This is due tothe fine material that composes the shale being deposited inthe non-uniformities of topography in areas of weak turbu-lence. As a consequence the thickness of the deposits is relatedto the local topography prevailing at the time of depositionand is independent of depth.

Here, our approach in evaluating shale thickness is to definea broad qualitative classification of depositional environmentsagainst thickness (Fig. 11). This is based on the principles ofsediment deposition guided by knowledge acquired from anunderstanding of the outcrop geology and sedimentary archi-tecture. Shales represent deposition under lower energy condi-tions than the sandstones and may relate to either lower energy

locations within the depositional system or periods of lowerenergy. Sediments deposited in different depositional environ-ments generate different patterns of sandstone and shale andthe degree of preservation of shale within sandstone domi-nated intervals will vary from environment to environment. Asudden change in process or energy may allow the temporaryaccumulation of some thin shales. Thus, the top end membersof Fig. 11 are those reservoirs that are commonly dominatedby shales as their depositional processes are mostly low en-ergy. For example, deep-sea slopes have thick sections of shalewith almost no sand. Shale is transported to these areas easily,whereas sand is regarded as being a more localized input spa-tially and temporally. Indeed, even the sand-prone portionsare likely to include some zones with thick shale – Fig. 12(a)gives an outcrop example of such a system. Systems that de-posit essentially sheet-like sandstones (such as unconstraineddeepwater turbidite systems) often produce interbedded sand-stones and shales, so that extensive shale-dominated inter-vals may be preserved within the reservoir. For the low endmember environments, the reverse is true – they are generallysand and gravel dominated, characterized by erosion as wellas deposition, with most finer grained sediment washed orblown away by high energy processes. For example, shalesare very rare in the aeolian systems (Fig. 12b), where gener-ally only thin playa lake shales exist. It is usually the fluvialincursions that will generate the shales. Also, in fluvial sys-tems individual channel sandstones within a shale-dominatedfloodplain succession may stack vertically and horizontally

Figure 11 Order of magnitude estimate of intra-reservoir shale thickness range for various depositional environments. Note that these are forparts of the depositional system in which only sands dominate. Values are defined for the most typical (frequently occurring) shale units. Drawnfrom contributions by Dorrik Stow (personal communication). Field names are shown for reference purposes. The acronym SGB stands for thesouthern gas basin.



Figure 12 Outcrop photographs of two depositional systems from our hierarchy of Fig. 11. Preserved shale thickness can be clearly seen in thedeep-sea fan/basin plain example of a) from the Annot sandstone, SE France but is mostly absent in the Aeolian dune sandstones of b) from theNavajo sandstone, Zion national Park, US – highlighted by the white arrows in both cases. Photographs are sourced from Dorrik Stow (pers.comm.).

to form thicker and more extensive sand-dominated sheets.Discontinuous shale bodies may then be preserved within thesandstone dominated channel complex and their shapes willbe irregular, reflecting the shape, size and stacking pattern ofthe channel sandstones. In other cases, the shape of the shalespreserved within a sandstone dominated interval reflects thesub-environment in which they were deposited. For example,in a braided river environment, sections of a river channel

may be abandoned and may become filled by shale. In thiscase, the preserved shales will reflect the shape and size of thechannels and will tend to be elongated parallel to the channelsandstones.

The different clastic depositional environments above canprovide the necessary range of shale dimensions anticipatedand examined in this study. Thus, for example, in reservoirscreated by the deep-sea slope environment such as in the



Girassol field (Navarre et al. 2002) or the Schiehallion field(Leach et al. 1999), the shales may be sufficiently thick (5–10m) to have the initial geomechanical extension offset overperiods of 1000 s of days by the pressure diffusion effects.When frequent time-lapse surveying of a 12-month survey pe-riod or less is carried out, the sub-seismic shales may thusbe in a partially equilibrated state and the reservoir softeningdiscussed above could be present in low net-to-gross regionsof the reservoir. For a longer time between surveys, reservoirhardening may prevail. However for other depositional envi-ronments such as the Heidrun field (Garn formation, Furreet al. 2006) or the Southern Gas Basin (MacBeth, Stammeijerand Omerod 2006), the shales are thin enough to experiencean initial geomechanical effect that is then rapidly cancelledover tens of days by the relatively rapid pressure diffusioneffects. Thus, surveys taken over a few years will exhibit theexpected reservoir hardening due to pressure depletion.

CONCLUSIONS

Sub-seismic, intra-reservoir shales of 1–10 m thick can af-fect the time-lapsed effective seismic response of the reservoir.Coupled pressure diffusion and geomechanical effects actingon these shales have an influence on the apparent in situ stresssensitivity of the reservoir and act to reduce the sensitivityfrom that expected by applying laboratory measurements car-ried out on predominantly sandstone core plugs. This effectadds to the list of factors contributing to a modification of thestress sensitivity when applied to the in situ reservoir state. Un-der certain conditions (low net-to-gross, frequently repeatedseismic, thick shales and low shale permeability) the reduc-tion in stress sensitivity anticipated from our modelling mayin fact reverse the polarity of the usually expected response,causing an apparent softening instead of hardening of thereservoir impedance with depletion. It is shown, for exam-ple, that seismic surveys repeated over 5–10 years are likelyto be influenced by 5–10 m thick shales such as those foundin low energy depositional systems, whilst thinner shales willnot present a problem for these surveys as they will pressureequilibrate over that time period. However, frequent seismicmonitoring on a 3–12 monthly basis will be influenced by sub-seismic shales over the entire thickness range of 1–10 m andthis is likely to be apparent for a wide range of depositionalenvironments.

The results of this work are strongly controlled by the cou-pled mechanical, transport and elastic properties of the shales.Such properties are notoriously difficult to assign due to agenerally insufficient knowledge and sampling of these rocks.

Additionally, such sub-seismic intra-reservoir shales may bedifficult to precisely identify within the inter-well volume, par-ticularly if they are truly ‘stochastic’ in nature. Thus the resultsmust currently carry an uncertainty that cannot be adequatelycalibrated in practice – this is why we have considered in thispaper simplified geology models but with realistic rock prop-erties and the correct physical mechanisms. By contrast, thecontribution from shales with more correlation and continuitybetween wells is more predictable as the shale distribution isbetter controlled. Some control over the expected reduction instress sensitivity can be obtained from geological interpreta-tion of outcrops, log and core data together with geomechan-ical and elastic wave stress sensitivity property measurements.Detailed facies analysis from 3D geological models could alsohelp to calibrate the predictions made in this study, by con-straining the shale size distributions and helping modellingstudies to estimate the arching factors and pressure equilibra-tion effects. The results of this work strongly indicate thatmeso-scale geology is essential to adequately assign elasticwave stress sensitivity to the reservoir.

Industry is currently doing very little about shales. Treatedas barriers to flow, they are generally assumed to have zeropermeability for the purpose of predicting the flow rate atwells and fluid flow in the reservoir. However, is it adequate torepresent shales simply as barriers in fluid flow calculations orshould they be given their own distinct permeability, capillarypressure and mechanical properties? Shale mechanical prop-erties are particularly important for drilling but their abilityto alter the seismic scale response also needs to be addressedand may in turn feed back into monitoring of the drilling pro-gramme. Our work suggests that the need for shale propertiesshould be recognized and measurement become more com-mon practice. For this to occur, an accurate assignment ofreservoir shale properties is a topic that requires much morefuture attention.

ACKNOWLEDGEMENTS

This work was sponsored by the Edinburgh Time-LapseProject, Phase III and is published with approval from itssponsors: BP, BG, Chevron, ConocoPhillips, EnCana, Exxon-Mobil, Hess, Ikon Science, Landmark, Maersk, MarathonOil, Norsar, Norsk-Hydro, Petrobras, Shell, Statoil, Totaland Woodside. The authors thank Dorrick Stow and PatrickCorbett of IPE HWU for constructive discussions on the as-pects on reservoir geology. Ludovic Ricard is acknowledgedfor his contribution to the pressure diffusion aspects of thiswork. Peter Schutjens is acknowledged for his discussion on



technical aspects related to this work. The authors acknowl-edge the thoughtful comments made by the reviewers PeterSchutjens and Joel Sarout, which have helped to enhance thesubmitted manuscript.

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APPENDIX A: COMPUTATION OF THESTRESS ARCHI N G R A T I OS γ sh AND γ sa

Prior to production, the reservoir and its surrounding rockare in mechanical equilibrium, with the total externally ap-plied stresses matching the internal reservoir stresses. Afterextraction of a fluid volume during production, the fluidpressure in the reservoir changes by �P and this induces atotal internal stress change (both vertical and horizontal) of�σtot(int) = αsa�P (where αsa is the effective stress coeffi-cient) that consequently creates a mechanical imbalance. Thereservoir and surrounding rocks react to this imbalance bystraining, the mechanics of which are governed by the totalexternal and internal stresses and the boundary conditions –for laterally extensive reservoirs this strain is mostly directedin the vertical direction. This change of reservoir geometryredistributes the total external confining stress. The degree ofredistribution depends on the magnitude of the causative in-ternal stress change, which depends in turn on the changeof reservoir pressure and the volume extracted. FollowingHettema et al. (2000), the change in the total confining verticalstress �σ tot

v induced by this pressure change can be written as�σ tot

v (ext) = γ v�P, where γ v is a positive coefficient. Thus,the resultant change in the effective stress loading the rockframe of the reservoir is �σ

effv = (γ v − αsa)�P. It is this ef-

fective stress that is important when determining the materialproperties of the rock in our calculation in the main text asillustrated by Fig. 3. A similar change and definition exists inthe horizontal direction, for which γ h is defined. For the mod-

els considered in this current work it is found that γ h > γ v.Both γ v and γ h are non-negligible in our particular case studydue to the geometry of the reservoir and thus unequal velocitychanges due to the vertical and horizontal total stress changesare expected. However, as our work focuses only on the im-pact of vertical velocities and these velocities are affected onlyby the vertical stress changes (assuming material isotropy andhomogeneity), variations in γ h are ignored. Thus, in the sub-sequent analysis and in the main text γ v will be referred to asγ − either γ sh for shale and γ sa for sand.

The work below considers the influence of intra-reservoirshales on the stress arching ratios for both the sands γ sa andshales γ sh. We consider the geomechanical behaviour due to asingle sand body containing layers of sand and shale beds re-sponding to production-induced depletion. Depletion is onlyin the sands as the intra-reservoir shales cannot be produceddue to their very low permeability, however they do inter-act mechanically due to the stress imbalance created by thesurrounding sands. The γ sh values depend primarily on thethickness and distribution of the beds in the channel andthe channel geometry and dimensions. To determine γ sh forthe purpose of the current study, 3D numerical modellingof the mechanics of depletion is carried out for a range ofchannel models containing a distribution of thin shale beds.Each channel cross-section is assumed to be semi-elliptical andis extended in the transverse direction (Fig. A1). The channelsconsist of alternating layers of thicker sandstone beds sepa-rated by shale eroded by various degrees. The sand quality isassumed to be highest at the base and the centre of the chan-nel and to decline upwards and outwards (Stephen, Clark andGardiner 2001). For the purpose of this modelling, the chan-nel body is discretized into a number of orthogonal cells withdimensions 0.5 m × 0.5 m × 0.5 m (Fig. A1). The reservoir

Figure A1 The channel sand model is discretized into cubes of 0.5 m in dimension. Calculations based on the equations of Geertsma (1973)and described in Appendix A are performed for a central vertical slice through this model. Intra-channel shales are distributed according to theerosional model of Stephen et al. (2001). The lower right figure is the contoured net-to-gross in the channel, where red is low and blue is high.



deformation resulting from withdrawal of fluid from each in-dividual sand cell is calculated, the result summed for all sandcells in the reservoir and then the theory of linear poroelastic-ity used to calculate the impact on the effective stress and to-tal stress. The reservoir deformation calculation is performedusing the method of infinitesimal inclusions (the nucleus ofthe strain method of Geertsma 1973) as implemented by Hu(1989). This gives the displacement, Ui, at a specific pointdue to an infinitesimal inclusion subject to depletion-inducedstrain (in this case an individual sand cell) embedded in anisotropic homogeneous half-space. An assumption implicit inthe use of this approach is that the elastic constants for thesand and shale must be identical. The vertical and horizontaldisplacements at the cell-cell interfaces due to pressure changein the whole reservoir can thus be readily obtained as the su-perposition of the displacements due to each depleting sandcell in the reservoir. Having obtained the displacements ateach interface between the sand-sand cells or sand-shale cells,the cumulative strain εi j = 1

2 ( ∂Uj∂xi

+ ∂Ui∂xj

) on each shale bed orparticular portion of sand can be determined. The strain fieldin turn can be converted into the total stress perturbation�σ tot

ij by applying the theory of poroelasticity

�σ toti j = μ

(∂Ui

∂xj+ ∂Uj

∂xi

)+ 2μν

1 − 2ν

(∂Ui

∂xj+ ∂Uj

∂xi

)δi j

−α�Pδi j ,(A1)

where μ is the shear modulus, ν the Poisson’s ratio and δij

Kronecker’s delta (in our case μ = 25 GPa and ν is fixedto 0.25). After the stress change is known, the γ values canbe determined by dividing by the pressure drop �P. In thecase of the sand, the effective stress coefficient, αsa, is set tounity, whilst for the shale, αsh, is zero for the impermeablecase and 0.7 for the permeable case (once pressure diffusionhas begun – see Appendix B). The main text briefly explainsthe choice of these values.

The above calculation is carried out for four different sizesof channel with thickness: width ratios: 10 m × 50 m, 10 m ×100 m, 15 m × 250 m and 20 m × 500 m and five differ-ent thicknesses of shale bed (0.5 m, 1 m, 2 m and 10 m).It is performed in 3D but the results for only a vertical sec-tion in the centre of an 800 m long sand body are used –i.e., in the calculation we choose to ignore the calculationsat the channel edge. As anticipated, for each channel we findthat the deformation of the top and bottom of the channelsare in opposite direction, i.e., the roof subsides and the floorrises. Inside the reservoir, the displacements at the sand/shaleboundaries are also not identical and they all indicate shaledilation but by differing amounts. For each of the chosen mod-

els, the number of shales and hence the net-to-gross, is varied.The net-to-gross is also varied laterally across each sand chan-nel in agreement with the conclusions of Stephen et al. (2001).The results show that for the range of models considered, γ sa

is approximately equal to γ sh. Also, as the net-to-gross in-creases, so too does γ sh, (Fig. A2a). Thus, for example, γ sh,varies from 0.2–0.5 for a shale thickness of 1 m. In addition,γ sh, is greater for shales at the top of the channel than forshales near the base of the reservoir (where there is more de-pleting sand and thus more shale extension) (Fig. A2b). γ sh,also varies with the shale thickness and for the 1 m thickshales it lies between 0.1–0.4, whilst for 2 m thick shales therange reduces to between 0.1–0.2. For the 10 m thick casethe net-to-gross is very low in some of the channels, γ sh issmall, less than 0.1 (but still positive). Our four cases andtheir channel thickness to width ratio R give: 10 m × 50 m(R = 0.2, γ v = 0.25), 10 m × 100 m (R = 0.1, γ v = 0.15),15 m × 250 m (R = 0.06, γ v = 0.04) and 20 m × 500 m (R =0.04, γ v = 0.02). The values are similar to those reported inthe literature for a disc-shaped or ellipsoidal reservoir as mostof the changes are found to be governed by the thickness towidth ratio (Hu 1989). The results from the study above areused as input to the rock and fluid physics calculations in themain text.

APPENDIX B: PRESSSURE EQUIL IBRATIONIN A S HALE BED

Consider a shale layer sandwiched between two thick sandlayers produced via a single vertical well (Fig. B1). All layersare uniform and have constant fluid flow properties. In thismodel example, there is some aquifer support from below butno lateral injection support. The pressure drawdown fromthe well is regulated so that a constant pressure depletion ofbetween 5–10 MPa is established at an intermediate distanceaway from the well. During the initial stages of production atransient pressure behaviour is induced, after which pressureequilibrium is quickly attained in the sands and a gradient isestablished vertically across the shale. This vertical pressuregradient then drives pressure to equilibrate in the shale bed.Sand-shale equilibration occurs at a much slower rate thanthe depletion in the sands due to production because of thelower shale permeability – a matter of tens of days rather thanseconds.

An approximate solution for the time taken in the processof shale equilibration can be calculated analytically using the1D pressure diffusion equation. Assuming application to ahomogeneous porous media containing one or more slightly



Figure A2 Gamma values for shale, for a range of thicknesses and channel dimensions: a) as a function of net-to-gross in the channel and b)depth location in the channel. Channel depths have maximum limits from 2000 m (top reservoir) to 2020 m (base reservoir).

compressible fluid phases, the following equation is appropri-ate (Dake 2001)

∂ P∂t

= D∂2 P∂z2

, (B1)

where P is the desired pressure, t and z the time and ver-tical coordinate respectively. The diffusivity coefficient D =λ/φct accounts for the total (volume-weighted) compressibil-ity of the formation and fluids ct, the porosity φ and finallythe total fluid mobility λ. The mobility is given by kkr/μ –where k is the permeability, kr the relative permeability and μ

the fluid viscosity and the total mobility is the summation ofthese quantities for all of the fluid phases present in the rock.The diffusivity coefficient D is assumed for the purposes ofthis calculation to be constant with production. The equationis applied to the vertical pressure profile Psh(z, t) in the shaleand defined between z = 0 (base of shale) and z = L (topof shale) and for all t > 0 (Fig. B2). Equation (B1) is solvedsubject to a number of boundary conditions. Firstly, the initial

pressure inside the shale is uniform and equal to that of thesand: Psh(z, 0) = Psai for 0 ≤ z ≤ L. Secondly, compared tothe time scale of equilibration in the shale, the pressure in thesand is assumed to instantaneously deplete from Psai to Psaf

in response to production, this giving a periodic condition atthe top and base of the sand: Psh(0, t) = Psaf and Psh(L, t) =Psaf for all t > 0 and repeated every L. Equation (B1) can besolved to a reasonable degree of accuracy using the methodof separation of variables or more exactly (at early times) byLaplace transforms (see, for example, Crank 1975). The for-mer leads to the solution for the pressure in the shale as aseries of damped sinusoids

Psh(z, t) = Psa f + (Psai − Psa f )∞∑

n=0

bn exp(

− (2n + 1)2π2 DtL2

)

× sin(

(2n + 1)πzL

), (B2)

where bn = 4/(2n + 1)π . The resultant solution across theshale layer is evaluated numerically and plotted in Fig. B2.



Figure B1 3D numerical simulation of pressure evolution in a two-phase (oil and water) 2D reservoir model defined by a homogeneous sandwith an embedded central shale layer. The sand has a permeability of 500 mD and the shale is 1 μD. Porosity is 20% throughout. There is asingle production well of variable rate regulated to give 5–10 MPa depletion and there is some aquifer support from below the model.

Figure B2 Pressure profiles (in red), before (Psai), immediately after (Psaf ) and as time evolves in the depletion of the producing sands either sideof a low permeability shale layer. Pressure equilibration in the shale evolves slowly over a significant time scale (tens of days) compared to asimilar process applied to high permeability sands (seconds). The example here is for a 1 m thick shale of 1 nD permeability. It takes 82 days forthe shale to reach within 10% of the depleted pore pressure in the sands. Pressure solutions in the shale are shown for a number of intermediatetime steps and are created by summing the first 100 harmonics of the solution defined by equation (B2).



Table B1 Reservoir fluid and rock properties and their corresponding diffusivity coefficient, D and time constant2.3τ controlling pressure depletion in the shale. The shale is assumed to be 100% saturated with water. Whent = 2.3τ the pressure drop in the shales is 90% of that in the sands. Values are assigned using typical North Seareservoir conditions (MacBeth, Stephen and McInally 2005) as an exemplar and calculations of D and τ are for arange of absolute shale permeabilities and thicknesses. Permeability k; viscosity μw; rock compressibility cr; watercompressibility cw; and porosity as a percentage of bulk volume φ. L is the thickness of the shale layer

k μw cp Cr (1/psi) ×10−6 Cw (1/psi) ×10−6 φ D m2/day L m 2.3τ = 2.3L2/Dπ2 days

1 mD 0.38 3.0 3.5 0.10 2860 1.00 0.000081 μD 0.38 3.0 3.5 0.10 2.860 1.00 0.081 nD 0.38 3.0 3.5 0.10 0.002860 1.00 82

1 mD 0.38 3.0 3.5 0.10 2860 10.00 0.0081 μD 0.38 3.0 3.5 0.10 2.860 10.00 8.21 nD 0.38 3.0 3.5 0.10 0.002860 10.00 8149

The shale pressure decays exponentially as a function of timeand the rate of decay is controlled by the value of the diffu-sivity coefficient and also the square of shale thickness. Toobtain estimates for the time for this depletion process to oc-cur, typical reservoir rock and fluid parameters are assignedbased on a North Sea clastic reservoir (see Table B1), fromwhich D and the time constant τ = L2

π2 Dare determined. It is

expected that the shale will be predominantly water saturateddue to its high capillary entry pressure and therefore the dif-fusivity coefficient used in this calculation will remain fairlyconstant with pressure and temperature. (Note that the totalmobility and compressibility will vary strongly as a functionof reservoir saturation and pressure in the presence of gas).At later times the fundamental mode (n = 0 term) dominatesand the behaviour of this mode may be used to estimate thedepletion time. When the amplitude of this mode decays suchthat exp(−t/τ ) ≤ 0.1, the difference between Psh and the de-pleted state Psaf is considered negligible and the shale is os-tensibly equilibrated. The time, t = 2.3τ for this amplitude tobe reached is evaluated for different shale layer thicknesses L

and a range of absolute shale permeabilities k and these areplotted in Fig. 6 in the main text. The characteristic time forequilibration (t = 2.3τ ) for a 1 m thick shale layer of 1 nD

is roughly 82 days whereas for a 10 m thick layer of similarshale the process takes significantly longer at 8149 days (22years). In contrast, changing the permeability to 1 mD gives0.12 minutes and 12 minutes for 1 m and 10 m thick shales,respectively.

Note that the time constants for shale depletion indicatedin Fig. 6 are also expected to be applicable to the 2D and 3Dcase. In general, shale depletion in a layer of varying thick-ness will locally equilibrate by different amounts along thelayer and may to some degree be affected by pressure gra-dients internal to the shale. Additionally, if the shale hasinternal laminations of, say, silt or sand with a higher per-meability, these will deplete faster and the composite ‘shale’layer will be controlled by thicknesses of the individual lowpermeability pure shale layers. Furthermore, if the shale isdistributed as a discrete body totally embedded in the de-pleting sands, the depletion rate is controlled by the rangeof smallest thicknesses of that body. However, for the caseof a depleting sand and very thick overburden shale, solu-tion (B2) is not applicable and the treatment for a semi-infinite problem yields a different solution that accountsfor the depth of penetration into the shale (MacBeth et al.

2010).


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