rock physics modelling of 4d time-shifts and time-shift derivatives using well log data - a north...

Geophysical Prospecting, 2013, 61, 380–390 doi: 10.1111/j.1365-2478.2012.01134.x

Rock physics modelling of 4D time-shifts and time-shift derivativesusing well log data – a North Sea demonstration

Per Avseth1,2∗, Norunn Skjei3 and Ashild Skalnes3

1Norwegian University of Science and Technology, Trondheim, Norway, 2Spring Energy, Oslo, Norway, and 3Statoil ASA, Bergen, Norway

Received February 2012, revision accepted July 2012

ABSTRACTRock physics models for fluid and stress dependency in reservoir rocks are essential forquantification and interpretation of 4D seismic signatures during reservoir depletionand injection. For siliciclastic sandstone reservoirs, the Gassmann theory successfullypredicts changes in seismic properties associated with fluid changes. However, ourability to predict the sensitivity to pressure from first principles is poor, especially forcemented sandstones. In this study, we demonstrate how we can use a patchy cementrock physics model to quantify the combined effect of stress and fluid changes interms of seismic time-shifts and time-shift derivatives during depletion or injection.The time-shifts are estimated directly from well log data without core calibration ofstress sensitivity. By assuming non-uniform grain contacts where some grain contactsare cemented and others are loose, we can combine the contact theory for cementedsandstones with the contact theory for loose sands in order to predict stress sensitiv-ity in a patchy cemented sandstone reservoir. Time-shift derivatives are also usefulestimates, as this parameter reveals which part of the reservoir is most stress sensitiveand contributes most to the cumulative time-shift.

We test out our new approach on well log data from Troll East, North Sea andcompare the predicted time-shifts with observed 4D seismic time-shifts. We find thatthere are good agreements between predicted time-shifts and observed time-shifts.Furthermore, we confirm that there are local geological trends controlling the fluidand stress sensitivity of the reservoir sands on Troll East. In particular, we observea lateral stiffening of the reservoir from west to east, probably associated with thetectonic and burial history of the area. The combined effect of a thinning gas cap andstiffening reservoir sands amplifies the eastward decrease in time-shifts associatedwith reservoir depletion. We manage to disentangle these two effects using rockphysics analysis. It is essential to identify and map the static rock stiffness spatialtrends before interpreting time-shifts and time-shift derivatives in terms of dynamic(i.e., 4D) pressure and fluid changes.

Keywords: Time-shift, Rock physics, Stress sensitivity

INTRODUCTIO N

Proper analysis of 4D seismic signatures requires a goodunderstanding of rock physics properties as a function of

∗E-mail: [email protected]

fluid and stress changes. We know very well how to predictseismic sensitivity to fluid changes, using the Gassmann the-ory (Mavko et al. 2009). However, our ability to predict thesensitivity to pressure from first principles is poor. The currentstate of the art requires that we calibrate the pressure depen-dence of velocity with core measurements. A major challengeis the fact that consolidated rocks often break up during coring

380 C© 2013 European Association of Geoscientists & Engineers

Rock physics modelling of 4D time-shifts and time-shift derivatives 381

Figure 1. Schematic geologic cross-section of the Troll East fault block, showing target formations, fluid contact and two key wells that arefocused on in this study. There is a very thin oil leg observed beneath the gas cap in some of the wells, hence we sometimes denote the fluidcontact as a gas-oil contact (GOC) in stead of gas-water contact (GWC) in this study.

and hence the stress sensitivity is likely to be over-predictedin the laboratory relative to in situ conditions (e.g., Furreet al. 2009). Core samples are furthermore often selected fromthe best quality reservoir sandstones but are not necessarilyrepresentative of the complete reservoir. For unconsolidatedsands, acquisition of core samples is not very feasible due tothe friable nature of the sediments but the Hertz-Mindlin orWalton Smooth contact theory has been found to work well inpredicting the pressure sensitivity in loose sands (e.g., Zimmeret al. 2007; Bachrach and Avseth 2008).

In this study, we apply a patchy cemented sandstone modelby Avseth and Skjei (2011) to predict stress sensitivity ofpoorly to moderately consolidated sandstones. This modelquantifies stress sensitivity in cemented sandstones using anon-uniform contact theory combined with modified Hashin-Shtrikman elastic bounds and this ‘hybrid’ model will allowus to predict the pressure sensitivity in cemented sandstonesfrom well log data alone. We demonstrate how we can usethis approach to predict time-shifts and time-shift derivativesduring depletion or injection.

We test out the approach on real data from the Troll Eastfield, North Sea. Our rock physics based modelling results oftime-shifts are compared with measured 4D seismic time-shiftsin Troll East. We also investigate the effect of static reservoirproperties on dynamic reservoir properties. In particular, westudy the effect of burial history and associated variation inrock stiffness on fluid and stress sensitivity in the area. Figure 1shows a schematic cross-section of Troll East, where we have

highlighted the effect of differential compaction from westto east. Because of the structural dip and varying thicknessof the overburden, the Sognefjord Formation will be morecompacted in the east than in the west. Avseth and Dræge(2011) documented that diagenesis and cement volume arelarger in the east compared to the west in this field. However,time-shifts at the gas-water contact (GWC) will also reducefrom west to east due to a thinning gas cap. In this study,we use rock physics analysis to disentangle the effect of thethinning gas cap from the effect of stiffening reservoir rock,on 4D time-shifts. There are also facies changes in the TrollEast field that could exhibit varying stress and fluid sensitivity.However, in this study we assume clean sands (i.e., clay-free)within the reservoir. In spite of some variability in sandstonefacies, we find this to be a valid assumption as seen in theshale logs (Vsh) in Fig. 4.

THE PATCHY C EMENT R OCK PHYSICSMODEL

Avseth and Skjei (2011) suggested an approach to pre-dict elastic stress sensitivity in cemented sandstones using anon-uniform contact theory combined with modified Hashin-Shtrikman elastic bounds. This is an extension of the staticrock physics models presented by Avseth, Mukerji and Mavko(2005). It is assumed that the cemented rock will consist in abinary mixture of cemented and uncemented grain contacts, or‘patchy cementation’. Assuming that the cemented ‘stiff’ grain

C© 2013 European Association of Geoscientists & Engineers, Geophysical Prospecting, 61, 380–390

382 Per Avseth, Norunn Skjei and Ashild Skalnes

Figure 2. The patchy cement model combines the Hertz-Mindlin (i.e.,Walton smooth) contact theory and the Dvorkin-Nur contact cementmodel. It represents an extension of rock physics models for cementedsandstones originally made for diagnosing rock texture and staticreservoir properties to also take into account the effect of pressure(adapted from Avseth and Skjei 2011a). Data points falling on thesoft bound (light brown lines) are unconsolidated and will have pres-sure sensitivity according to the Walton smooth contact theory. Datapoints falling on the stiff bound (black lines) will have no stress sen-sitivity as all grain contacts are assumed to be cemented. Data pointsfalling in-between the soft and stiff bounds will have stress sensitivitygiven by a linear bounding averaging weight factor. The colour scalerepresents cement volume.

contacts are stress-insensitive and the unconsolidated ‘loose’grain contacts are stress-sensitive according to the Hertziancontact theory, this hybrid model will allow us to predict thepressure sensitivity in cemented sandstones (Fig. 2).

A weight function, W, is defined depending on where thesandstone data plot between an upper and lower bound in theelastic moduli versus porosity domain:

W = Kdry − Ksoft(P0)Kstiff − Ksoft(P0)

, (1)

where Kdry is the dry bulk modulus of a cemented sandstoneat a given porosity, Ksoft is the pressure sensitive soft (lowerbound) bulk modulus at the same porosity estimated at agiven reference pressure, P0 and Kstiff is the pressure insensi-tive stiff (upper bound) bulk modulus at this porosity value.Here we assume that the relative position between the boundsis linearly related to the pore stiffness and associated pressuresensitivity, c.f., the bounding average method introduced byMarion and Nur (1991). The soft bound is the unconsolidatedsand model where the reference effective stress (P0) is normallyset to 20 MPa in continuously subsiding basins. This repre-sents the effective stress at around 2 km burial depth, which

is the depth we expect initial quartz cementation to initiatein the North Sea. For uplifted rocks, we choose to use thein situ effective pressure as the soft bound pressure. Any datapoint falling on this soft bound should represent unconsoli-dated sands where all grain contacts are stress-sensitive. Thestiff bound represents the situation when all grain contacts areclosed and there is no stress sensitivity in the sandstone datafalling on this bound. A similar weight function is defined forthe shear modulus. These weight factors will define the stresssensitivity of the cemented sandstone and we obtain a modi-fied contact model for heterogeneous contacts that is pressuresensitive via the fraction of unconsolidated grain contacts:

Kdry(Peff

) = (1 − WK ) Ksoft(Peff ) + WK Kstiff , (2)

Gdry(Peff

) = (1 − WG) Gsoft(Peff ) + WG Gstiff . (3)

Figure 2 shows an example of simulated data using thepatchy cement model (Avseth and Skjei 2011), where we plotdry rock bulk modulus versus porosity and effective pressure.Note that the data plotting close to the lower bound shows sig-nificant pressure sensitivity, whereas the well cemented dataplotting close to the stiff bound show no or insignificant stresssensitivity. Some data points may fall below the defined softbound. We can either choose to define these as outliers andassign no pressure sensitivity to these, or (if they are fallingjust below the soft bound) give them a weight factor of zeromeaning these data points will have the same pressure sensitiv-ity as the soft bound. We often see that intra-reservoir shalesor shaly sands will plot below a clean sand soft bound. Forshales, it makes sense to assume no pressure sensitivity duringproduction. For shaly sands, we could establish an updatedsoft bound that takes into account clay volume in the rockframe.

The model presented here should be strictly valid only forrelatively large porosities, when grain contacts are controllingthe elastic stiffnesses. For relatively low porosity sandstonereservoirs, pore shapes and microcracks will be more im-portant factors controlling stress sensitivity. The modellingapproach presented by Shapiro (2003) or by Vernik andHamman (2009) should be more applicable to predict stresssensitivity in these types of reservoirs.

Using the rock physics model described above combinedwith the Gassmann theory, we can quantify the change invelocity as a function of pressure and fluid changes. The ve-locity versus effective stress is estimated from the dry rockmoduli and densities at each depth sample in the reservoirzone. This procedure can be summarized in two steps: thefirst step is to estimate the dry rock incompressibilities (Kdry)



Figure 3. Example of Vp (dry) versus differential stress estimated fora given target interval in a well in Troll East, using the patchy cementmodel approach. The colour represents probabilities (add up to 1 foreach pressure value). Fluid effects are later added using the Gassmanntheory.

and total porosities from the measured well log data (usingthe Gassmann theory and the mass balance equation), whichwill enable the estimation of the weight factor according toequation (1). The second step is to extrapolate the dry rockproperties at each depth sample, to any pressure using equa-tions (2) and (3). The effective pressure (Peff ) can be definedas a vector from let’s say 0–20 MPa, Hence, we can obtaina distribution of stress curves as shown in Fig. 3. Each depthsample in a given well included in the analysis (i.e., the reser-voir zone) will yield a separate stress curve in accordance towhere this sample or data point plots between the soft andstiff bounds in the porosity-moduli domain (c.f., Fig. 2). Su-perimposing stress curves for all the depth samples, we cangenerate probability density functions (pdfs) of velocity ver-sus effective (or differential) stress, as shown in Fig. 3. At eachpressure value, the probabilities add up to 1 along the velocityaxis (i.e., binning of individual stress curves is only done in thevelocity direction). The individual stress curves can further-more be used to estimate the velocity changes correspondingto depletion or injection related stress or fluid changes.

As mentioned above, the Gassmann theory is used to ac-count for fluid effects. The Batzle and Wang (1992) equationscould be used to estimate the fluid incompressibilities and den-sities at both in situ and monitor pressures. This is found to bea second-order effect compared to the pressure effect on therock frame. We find that the cumulative time-shift effects dueto the pressure effect on fluids at the GWC are less than 10%of the total time-shifts. Since fluid properties are also affectedby temperature changes and we do not have good control on

the temperature changes during depletion, we decided to ne-glect changes in fluid properties between the base and monitorin the examples shown in this paper. It is, however, easy toimplement pressure and temperature effects on the pore fluidsin our modelling using the Batzle and Wang equations.

Assuming hydrostatic pressure conditions, we can useTerzaghi’s principle to estimate in situ overburden (Plitho), ef-fective (Peff ) and pore pressures (Ppore). We assume the forma-tion factor, n, to be equal to 1, which is a good approximationfor poorly consolidated rocks.

Peff = Plitho − n Ppore. (4)

Then the effective pressure can be estimated by integratingthe density log for the whole overburden subtracting the fluidpressure. We neglect the seawater column since it will cancelout for hydrostatic conditions:

Peff =∫ Z=k

Z=0

(ρb − ρfluid

)gdz, (5)

where g is the gravitational acceleration.

TIME-SHIFTS A ND TIME-SHIFTD E R I V A T I V E S

The 4D time-shift within a reservoir zone can be estimatedfrom velocity changes associated with fluid and/or pressurechanges, according to the following equation:

�TWT = TWTmonitor − TWTbase

= 2 Z[

1

VP,monitor− 1

VP,base

]. (6)

(In this study, time-shifts are defined with the opposite sign,in accordance with the observed seismic time-shifts.) Time-shift derivatives, or time strains, are given by the followingequation (see also Landrø and Stammeijer 2004; Hodgsonet al. 2007; Rickett et al. 2007):

�TWTTWTbase

= �ZZbase

− �VP

VP,base. (7)

By assuming no change in macro-porosity (i.e., dilatation),the first term on the right-hand side in equation (7) will cancelout. For siliciclastic reservoirs, it is reasonable to assume nochange in macro-porosity during depletion.

Based on production related stress and fluid changes andassociated velocity changes we can derive the change in trav-eltime, according to equation (6). The time-shift estimatescan be estimated sample-by-sample for a given well. This willallow us to honour depth trends and geological variability



within a reservoir. The time-shift is then given by the integral:

�TWT(z) = 2∫ Z,bottom

Z,top

[1

VP,monitor (z)− 1

VP,base(z)

]dz. (8)

Then, the time-shift derivative is given by:

d (�TWT) /dt =2�Z

[1

Vp,monitor (z)− 1

VP,base(z)

]

2�ZVp,monitor (z)

= 1 − VP,monitor

VP,base= VP,base

VP,base− VP,monitor

VP,base

= �VP

VP,base. (9)

Hence, we see that the time-shift derivative, which is thevertical slope of the time-shift (as a function of time-depth), isequal to the fractional change in velocity for a given samplein the well log data. (Note that the time-shift derivative in thefigures is written as dT/T, denoting the change of TWT during4D, over the considered TWT vertical interval.) See Rickettet al. (2004) for more detailed explanations on time strains.

MODELLING T HE EFFECT OF PRESSUREAND FLUID CHA N GE S IN T R OL L E A ST

Next, we perform rock physics modelling of expected time-shifts and time-shift derivatives given the observed depletionpressures between the time of base and monitor surveys, forselected wells in the Troll East area.

The effective pressure increase during depletion is around40 bar in the Sognefjord Formation and around 20 bar in theFensfjord Formation. Figure 4 shows the associated time-shiftand time-shift derivatives predicted for two selected wells,Well A and Well B, respectively. For Well A, located in thewestern area of Troll East where we have a relatively thick gascolumn, the time-shifts are close to 2 ms at the gas-oil contact(GOC) (a thin oil leg is present beneath the gas cap), whereasthe time-shift derivatives show values averaging around 2%in the Sognefjord Formation and less than 1% in the FensfjordFormation. The variability in the time-shift derivatives prob-ably reflects the varying depositional units or facies withinthe two target formations. Different facies will have differentstress sensitivity.

Due to the decreasing thickness in the gas cap for Well B, aswe move from west to east on the Troll East fault block (seeFig. 1), the estimated cumulative time-shifts are lower at theGOC. This is due to a thinner gas cap but also due to the factthat gas saturated reservoirs will have larger stress sensitivity

Figure 4. Shale volume (Vsh) to the left, time-shift (dTWT in seconds)in the middle and time-shift derivatives (dT/T in fraction) to the right,for the two wells A (upper) and B (lower) in the Troll east area,associated with pressure depletion of around 40 bar in the SognefjordFormation and around 20 bar in the Sognefjord Formation. Note themuch larger time-shift at GOC and the base of the reservoir for WellA compared to Well B. The lower time-shift at GOC in Well B is dueto a thinner gas cap but also in general lower stress sensitivity in theSognefjord Formation in this well.

than brine-filled reservoirs when the rock stiffness is the same.Accordingly, the time-shifts at the GOC are lower in WellB compared to Well A. As we will document later, the east-ward decrease in time-shifts is also related to the compactionaltrends (i.e., burial history) in the area. Figure 5 shows pre-dicted time-shifts at six different well locations compared withobserved time-shifts. We observe a good quantitative matchbetween the predicted and observed time-shifts.

Due to the pore pressure drop during reservoir depletion,we expect gas coming out of the solution in the thin oil zoneand/or residual oil zone beneath the gas cap and perhaps alsoin the water zone. This effect will affect the time-shifts atlarger depths beneath the reservoir. As shown in Fig. 6, thereis a marked change in the time-shifts from west to east, goingfrom positive values in the west to negative values in the eastthat cannot be explained by pressure changes only.

In order to take into account the effect of gas out of so-lution on time-shifts, we use the Gassmann theory and fluidsubstitution in a given zone right beneath the gas cap. There



Figure 5. Predicted versus observed time-shifts at GOC for 9 wells in the Troll East area. Wells A and B are indicated. There is in general agood match between modelled and observed time-shifts. Observed time-shifts are courtesy of Statoil.

Figure 6. Time-shifts observed at 2100 ms, beneath the reservoir,showing a polarity change from positive values in the west (red colour)where Well A is located, to negative values in the east (black colour)where Well B is located.

are great uncertainties when it comes to saturation and thick-ness of this zone. Figure 7 shows the predicted time-shiftsand time-shift derivatives associated with combined pressurechanges and gas-out-of-solution for Wells A and B, where weassume residual gas saturation of 1– 2%, respectively withina 20 m thick zone below the GOC. Estimates of residual

gas in the neighbouring Troll West field indicated closer to1% residual gas below the GOC during depletion (IngvarSkaar, personal communication). For Troll East, we need toassume gas saturations of 2% within a 20 m zone, in or-der to obtain a qualitative agreement with the observed time-shifts in Fig. 6, where we have positive time-shifts at BaseFensfjord in Well 1 and negative time-shifts at Base FensfjordWell 2.

In the previous calculations we assumed no movement of thegas-water contact. However, there are reported local aquiferlifts in Troll East, in some areas as much as 30–40 m, basedon gravimetry data. In particular, relatively large movementshave been reported locally in the south-west corner of TrollEast. This effect can easily be incorporated into rock physicsmodelling using the Gassmann theory combined with assumedchanges in saturation profiles but is beyond the scope of thismodelling study.

LOCAL GEOLOGIC TRENDS A ND IMPACTON 4D TIME-SHIFTS

Avseth and Dræge (2011) documented that the Troll Eastreservoir rocks show significant lateral and vertical changes inseismic properties associated with compactional trends. Thisis attributed to the complex tectonic and burial history inthe area, where the pre-rift Jurassic reservoir rocks have beenfaulted and rotated, buried and uplifted (e.g., Fossen et al.2003). The absolute amount of uplift and timing of the events



Figure 7. Updated time-shift and time-shift derivative estimates for Wells A and B after accounting for gas-out-of-solution in the 20 m zonebeneath the gas cap. Assuming 2% gas saturation will still result in a positive time-shift at Base Fensfjord in Well A, whereas a change to anegative time-shift will occur at Base Fensfjord in Well B. This scenario can explain the observed time-shifts shown in Fig. 6. However, thereare great uncertainties in the combination of thickness and saturation of the zone where we model gas-out-of-solution and no unique answer tothis combination exists.



Figure 8. Seismic cross-section from west to east across Troll East.Note the eastward thinning gas cap, whereas the Cretaceous overbur-den is becoming thicker eastward.

are poorly documented. The extent of diagenesis of the sand-stones indicates that the Troll field could have been buried asmuch as 500–1000 m deeper than at present. (Horstad andLarter 1997).

Figure 8 shows a seismic cross-section from west to eastacross the Troll East fault block. We see that the gas columndecreases in thickness from west to east, whereas the Creta-ceous overburden increases in thickness in the same direction.In this section we will try to disentangle the effects of varyinggas column thickness and differential compaction related tothe burial history.

In Fig. 9 we have plotted predicted time-shifts at 100 mbelow the Top Sognefjord for the various wells in this study,for 100% water saturated reservoir sands and completely dryrock (i.e., 0% fluid), respectively (using the Gassmann the-ory). This figure should eliminate the effect of varying gascap thickness. As we see, the dry rock yields larger time-shiftsthan the water saturated reservoir rocks. However, we alsosee that both dry rock and water saturated rock show an east-ward decreasing trend in time-shifts. This must be related tothe rock stiffness variation and presumably this reflects the lat-eral variation in burial history discussed above and indicatedin Fig. 8.

In Fig. 9 we also superimposed the time-shifts for the in situ

saturation. We observe that for the western wells, the in situ

saturation will be commercial gas saturation down to 100 mbelow Top Sognefjord, hence the predicted time-shifts plotclose or onto the dry rock values. For the eastern wells, the

Figure 9. Cross-plot of time-shifts estimated at 100 m below the TopSognefjord Formation for 9 wells in the area. Blue circles represent thefully water saturated case, whereas grey squares represent dry rock.There is an eastward trend of decreasing time-shifts. We superimposedthe predicted time-shifts for in situ saturation (yellow circles). In thewestern wells the in situ saturation will be commercial gas saturationdown to 100 m below Top Sognefjord, hence the predicted time-shifts plot close or onto the dry rock values. For the eastern wells,the gas cap is much thinner or absent and the predicted time-shiftsplot close or onto the wet rock values. The west to eastward trendof decreasing time-shifts is hence amplified by the superimposition ofthe rock stiffness and gas cap thickness effects.

gas cap is much thinner or absent and the predicted time-shifts plot close or onto the wet rock values. The eastwardtrend of decreasing time-shifts is hence amplified by the su-perimposition of the rock stiffness and gas cap thickness ef-fects. The rock stiffness effect is further confirmed in Fig. 10,where we plot average shear modulus (within the 100 mupper Sognefjord interval) versus time-shift for the variouswells.

The exercise above demonstrates how important it is to un-derstand the local geology before we interpret the 4D seismicsignatures in a field. The different stress sensitivity that we ob-serve from east to west could easily be interpreted as differentpressure and/or fluid changes if we assume the same reservoirsandstone all over the field.

D I S C U S S I O N S

The patchy cement model used in this study assumes a binarymixture of grain contacts that are either loose (i.e., unconsoli-dated) or contact cemented. This differs from the assumptionnormally made in effective medium models using the contacttheory. Avseth et al. (2010) used a constant cement model to



Figure 10. Shear modulus versus predicted time-shift for the variouswells. Blue circles = water saturated rock; grey squares = dry rock.Note that in general the time-shift increases with reducing shear mod-ulus, confirming the effect of rock stiffness on time-shift variation.

quantify cement volume assuming that all grain contacts havethe same cement volume. The update to the binary mixtureis probably more reasonable in poorly to moderately con-solidated sandstones where initial cementation is occurring.The microstructure of sandstones is always more heteroge-neous than what we can account for in idealized rock physicsmodels. By assuming a binary mixture of grain contacts, wehonour some of this expected heterogeneity at the microscale.Bachrach and Avseth (2008) assumed non-uniform contactsand a binary mixture of grain contacts with no friction andno slip in loose sands and obtained a better match betweenmeasured velocities and predicted velocities using the con-tact theory. In a similar way, we can also consider the casewhere we have a binary mixture of loose and cemented graincontacts. The reason why some grain contacts are loose andothers are cemented can reflect local variability in grain shape,mineralogy, permeability or other parameters that affect thecementation process. By introducing a patchy cement model,we also allow for modelling of pressure sensitivity in cementedsandstones, which we know from in situ observations willbe present. The existing contact theory for contact cement(i.e., Dvorkin-Nur model) does not include pressure as a pa-rameter, as it assumes all grain contacts are uniformly ce-mented. The patchy cement model presented in this paperopens up for estimation of stress sensitivity in sandstones thatare cemented but where some grain contacts are still pressuresensitive.

The modelling approach applied in this study gives a goodmatch with observed time-shifts. However, as with any model,there are limitations and uncertainties. In this study we havehonoured heterogeneities in a microstructure but assumed100% quartz mineral in the solid phase. We know that thereservoir sands in the Troll East area locally have an abun-dance of mica, clay and calcite that has not been accountedfor in the modelling. Mineralogy variation can be accountedfor in the patchy cement model but without detailed mineral-ogy information in the well log data, it was not practical toaccount for this in this study. Furthermore, we have assumedno contact movement in the time-shift estimates shown in thisstudy but ongoing research is taking this effect into accountwhere it is found relevant in accordance with the informationfrom the reservoir simulation.

The patchy cement model use a linear weight between asoft, unconsolidated bound and a stiff non-compressible ce-mented bound, in order to quantify the stress sensitivity ofpatchy cemented sandstone. Recent research has shown thatthere is reduced tangential stiffness in loose sands associatedwith non-uniform grain contacts (Bachrach and Avseth 2008),varying internal friction at grain contacts (Duffaut, Landrøand Sollie 2010) or a heterogeneous stress pattern (Sain, Muk-erji and Mavko 2011). Any of these factors, separately or incombination, can explain why many have reported that thepressure dependency of elastic moduli, which arises from thecontact theory, does not always match the observed pressuredependence of moduli in unconsolidated material. A ‘reducedshear factor’ is applied to account for this and is found to beclose to zero at low effective pressures, in accordance with theWalton smooth contact theory formulation with zero friction.This parameter is found to increase with depth and increasingpressure and if it approaches one, we will have the Hertz-Mindlin or Walton rough contact theory formulation. Forany given field, like Troll East, this parameter will representan uncertainty in the modelling, which will have an impact onthe estimation of the weight factor that quantifies the stresssensitivity. This uncertainty will again propagate into the time-shift estimates. The stiff bound also has some uncertaintiesassociated with it. We do not know for sure what cement vol-ume will yield no stress sensitivity in the reservoir sandstones.We have assumed this to be around 10%, at which stage thecontact cement is assumed to become only pore-filling andthere is no more soft crack-like porosity to fill in between thegrain contacts. The linear weight is also an assumption thatmay have some uncertainty but Hashin-Shtrikman modellingof cemented grains mixed with loose grains in a connectedpatchy cement pattern has confirmed that a linear weight is



realistic (Avseth, Skjei and Mavko 2012b). Future researchshould focus on a sensitivity study of the various assumed pa-rameters of the bounds and investigate how these will prop-agate into the time-shift estimates. An improved physical un-derstanding of how we make effective rock physics modelsthat handle pressure sensitivity and how we do upscaling ofpressure is also required. Pressure sensitivity caused by large-scale fractures was not included in this study and could havesome impact on the observed seismic time-shifts. Nevertheless,the good match between modelled and observed time-shifts inthe Troll East area serve as a good validation of the approachdemonstrated in this study. The patchy cement model was fur-thermore validated by comparison with real data by Avsethand Skjei (2011), Duffaut, Avseth and Landrø (2011) andAvseth et al. (2012a).

CONCLUSIONS

We applied the patchy cement rock physics model by Avsethand Skjei (2011) to model the time-shift and time-shift deriva-tives due to pressure changes. The model requires only well logdata (Vp, Vs and density) and information about fluid proper-ties and in situ pressure and temperatures as input data. Themodelling approach enables us to efficiently perform com-prehensive rock physics modelling of different 4D scenarios,including pressure changes, gas-out-of-solution and contactmovements. In this study we focused on pressure changesand gas-out-of-solution associated with depletion and demon-strated the approach on well log data in the Troll East fieldin the North Sea. Comparisons between predicted time-shiftsfrom well log data and observed seismic time-shifts show agood match. Furthermore, we demonstrated that spatial vari-ability associated with facies changes and compactional trendswill affect the stress sensitivity of the reservoir rocks. Our stud-ies show that there is larger stress sensitivity in the westernpart of Troll East, compared to the eastern part of the field.It is important to honour these geological trends before inter-preting the time-shift attributes in terms of fluid and pressurechanges.

ACKNOWLEDG E ME N T S

We thank Anders Dræge, Havard Alnes, Harald Flesche andIngvar Skaar at Statoil and Frank Hauge at Odin Petroleumfor valuable discussions during this study. We acknowledgeStatoil and licence partners Norske Shell, ConocoPhillipsSkandinavia, Petoro and Total E&P Norge for allowing usto publish the data used in this study.

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rock physics modelling of 4d time-shifts and time-shift derivatives using well log data - a north...

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