Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. Vol.30, No.7, pp. 1193-1200, 1993 Printed in Great Britain

Numerical Simulation of Shear-induced Compaction in the Ekofisk Reservoir

0148-9062/93 $6.00 + 0.00 Pergamon Press Ltd

LEE CHIN* R. R. BOADE* J. H. PREVOST** G. H. LANDA***

Early efforts to simulate the reservoir compaction and subsidence phenomena for the Ekofisk oil and gas field (located in the Central Graben of the Norwegian North Sea) by usingfinite-element numerical methods, with pore collapse as the sole mechanism for compaction of the reservoir chalk, were quite successful in that agreement between measured and calculated subsidence magnitudes was good. In time, however, measured subsidence rates tended to be higher than calculated rates. This observation, coupled with field data that indicated deviatoric stresses in the reservoir had become very large, led to a decision to incorporate an additional compaction mechanism (referred to here as shear- induced compaction) into the compaction/subsidence simulation model for Ekofisk. The present paper describes the effort to develop and implement a constitutive formulation to treat this compaction mechanism in the reservoir compaction/subsidence model for Ekofisk.

INTRODUCTION

The work that has been in progress since late 1984/early 1985 to develop numerical methods for simulating reservoir compaction and subsidence at the Ekofisk field in the Norwegian sector of the North Sea has resulted in several progressively improved computational procedures [1-5], all of which used the DYNAFLOW finite-element code [6], and each, at the time of development, appeared to quite adequately describe the relevant physical processes. In time, however, after a few years of continued compaction and subsidence in the field, each of the procedures showed signs of weakness, suggesting there may be one or more physical mechanisms for compaction and/or subsidence that are active in the field but have not been properly taken into account in the computational models. The primary mechanism for compaction of the chalk reservoir rock that had been incorporated into all of the early simulation models was pore collapse, a process that in the high-porosity reservoir chalk at Ekofisk results in a marked decrease in pore volume as a result of an increase in effective stress; the increase in effective stress in the reservoir is caused by pressure decline, which in turn is caused by the removal of fluids from the reservoir by production. For the reservoir compaction and subsidence simulation models, reservoir

* Phillips Petroleum Company, BartlesviUe, OK ** Princeton University, Princeton, NJ *** Phillips Petroleum Company Norway, Tanager, Norway

pressure, based on information from the reservoir simulator for Ekofisk [7], has been used as basic input to quantify the compaction process.

During the latter half of the 1980's, the rate of decline of reservoir pore-pressure at Ekofisk slowed appreciably. In fact, for some portions of the Ekofisk reservoir, there have been periods of pressure increase caused by a water injection operation that was initiated in late 1987. In general, the average pressure decline rate during the 1988 through 1990 time period is about one-third of the decline rate during the 1985 through 1987 time period, and both of these average pressure decline rates are much lower than those realized during the late 1970's and ea'rly 1980'8. Thus, computational procedures that rely on pressure decline to quantify compaction, e.g., by pore-collapse, will predict a marked decrease in compaction and subsidence rates in response to a marked decrease in pressure-decline rates, unless there is a mechanism for compaction that yields an increase in the incremental compaction that results from a unit of pore pressure decline.

While subsidence rates did decline during a portion of the measurement period since 1985, the observed decline was not as much as predicted. It is this observation that led to speculation that one or more mechanisms for compaction, besides pore collapse, was active in the Ekofisk reservoir and to a concerted effort to identify that mechanism or those mechanisms. A portion of that effort is reviewed in this paper. The paper deals primarily with finite-element numerical simulation of the reservoir compaction and subsidence

1193

1194 R O C K M E C H A N I C S IN T H E 1990s

phenomena at Ekofisk, and the simulator that has been used for all calculation is the full'field, 3-D model that was developed in 1989 and 1990 [5].

M E C H A N I S M S FOR RESERVOIR COMPACTION AND SUBSIDENCE

There are a variety of physical mechanisms and/or processes that can and do contribute to reservoir compaction and subsidence; some of them pertain to the reservoir rock and some to the overburden. In paper attention will be given primarily to the mechanical behavior of the reservoir rock, although a few comments will be made about methods currently used to describe the behavior of the overburden. With regard to the reservoir rock, the discussion will be directed at the pore-collapse compaction mechanism and the mechanism we have termed "shear-induced compaction." The concept of shear-induced compaction stems from relatively recent observations about the evolution of in-situ stresses in the reservoir [8], namely that deviatoric stresses have become very large during the production history of the field, and from a conjecture that these observations have a connection to the fact that measured subsidence rates have tended to be higher than predicted rates during recent years.

Pore-collapse Compaction of Reservoir Rock Figure 1 shows composite stress-strain curves for

Ekofisk reservoir chalk that have been determined from numerous uniaxial-strain~ triaxial-stress compaction tests performed on intact cylindrical specimens, primarily

during the first two or three years after the discovery of subsidence at Ekoftsk in late 1984. Information is shown on the two graphs for the low-quartz-content and hlgh-quartz-content chalk~ found in the Ekofisk reservoir and in each case for chalks with even values of porosities from 30% to 40%. The Ekofisk field has two reservoirs, one in the Tor formation and the other in the overlying Ekofisk formation. High-quartz-content chalk is found in the Upper Ekofisk formation and low- quartz-content chalk is found in the Lower Ekoftsk and Tor formations. Porosity values represented in Figure 1 correspond to those used in the DYNAFLOW subsidence models.

Two stress-strain curves are shown for each material, one representing the strain experienced in a typical laboratory test and the other the strain that would have been experienced had the tests been conducted slowly enough to permit time-dependent strain (creep) to have been completed. These "ultimate-strain" stress-strain curves have been used in the models for simulating reservoir compaction and subsidence due to the pore- collapse mechanism as a means to eliminate the need to account for creep in the compaction/subsidence model.

Shear-Induced Compaction of Reservoir Rock After making measurements of in-situ stress in the

Ekoftsk reservoir for several years using the anelastic strain recovery (ASR) method, and following this work with a detailed analysis of pressure-time records from hydraulic fracturing stimulation operations, the data available have made it possible to plot effective horizontal stress versus effective vertical stress [8]. The

PO.osrry (%) 30 32 34 36 38 40

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a. ¢ ,," 40 ,. - - -

= IlgY/ 3O ¢n

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u_ 10 , . LU

H I G H Q U A R T Z C O N C E N T R A T I O N

poRourv (~) 30 32 34 36 38

0 [ - I I ! I I I 2 4 6 8 10 12 " 0 2 4 6 8 10 1Z 1 4 o

AXIAL STPJUN (%) AX~L STRAIN (%)

40 I

t

Fig. 1. Stress-strain curves illustrating the pore-collapse phenomenon in high and low quartz content reservoir chalk.

ROCK MECHANICS IN THE 1990s 1195

plot indicates that the slope of the data tends to be much shallower than expected; specifically, the slope (called K, sometime K o) is only about 0.2, instead of an expected value of about 0.5 based on uniaxial-strain, triaxial-stress compaction tests performed in the laboratory.

The significance of the low value for K is that as production has proceeded in the field, causing the pore pressure to decline, deviatoric stresses have increased by significant amounts. In fact, using laboratory data to estimate the envelope for brittle failure of chalk, it appears that shear stresses have developed sufficiently in the reservoir to cause fracturing of the rock by the early 1980's, maybe earlier. This is illustrated on Figure 2, which contains two Molar diagrams, one of which shows the development of deviatoric stresses (through the growth of Mohr circles) for a K value of 0.2 and the other for a K value of 0.5. It is clear that the growth of the Molar circle is much greater with a K value of 0.2 than with the larger value. By a similar analysis, it can be concluded that in-situ stresses in the reservoir have developed to the point that slippage on existing or newly created fractures will also occur.

02 u)

gc

cn

25

20

15

10

26

PORE 20 PRESSURE ~.

CURVE (psi) {MPa)

A 7000 48.3 ~ 15 B 6000 41.4

tn lO C 5000 34.5 K=0 .5 ~ / D 4200 29.0

~ ~'I~,~ ~ E 3500 24.1 ~ " ~ % ~ " F 3000 20.7

~ J ' ~ G 2600 17.9

5 1 0 1 5 20 25 3 0 3 5 4 0 4 5

NORMAL STRESS (PSI)

A.

¢n

u)

PORE PRESSURE

CURVE (psi) (MPa)

A 7000 48.3 25 [- / B 6000 41.4

/ K = 0 2 / C 5000 34.5 / *~- J D 4200 29.0

20 }- ,.t~.~'~ f E 3500 24. 1 / ,~,~, ~ ,~ * " F 3000 20.7 / ~q~7, '~ G 2600 17.9

B C D E F G 0 I I i i ii I I H I I I I I i i t i i I I I I

0 6 10 15 Z0 25 30 36 40 46

NORMAL STRESS (PSI]

Fig. 2. Mohr diagram for reservoir chalk illustrating the effect of declining pore pressure for cases where K = 0 . 2 a n d K =

0 . 5 .

In addition to the conclusion that fracturing and/or slippage on fractures will be induced by shear during pressure depletion, it is also quite likely that such behavior may also be induced by repressurization. This is illustrated on Figure 3, again for K values of 0.2 and 0.5 during depletion. The pressure increase phase is assumed to have been initiated at a time when the reservoir pore pressure had been reduced to 3500 psi (24 MPa), which is approximately consistent with the pressure existing when the waterflood was initiated at Ekofisk. The assumption has been made for this illustration that during repressurization horizontal and vertical effective stresses are decreased by the same increment, which is equal to the incremental increase in pressure. The key point to be drawn from Figure 3 is that repressurization forces the Molar circle to the left on the diagram, severely into the failure region for the K = 0.2 case and only marginally for the K = 0.5 ease.

PORE PRESSURE

CURVE (psi) (Mea) A 3600 24.1

K=0.6 DURING DRAWDOWN S 4000 27.8

K = 1' 0 DURING R E P R E S S U R I Z A T I O N ~ ~ CD S000450031"034. 6

. ~ ~ 0 ~ . ~ 41.4

i J 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5

NORMAL STRESS (MPa)

PORE PRESSURE

2 s t - K = 0 . 2 DURING DRAWDOWN . CURVE {psi) (MPa) / K = 1.0 DURING , ~ , ~ , , ~ A 3500 24. I [ REPRESSURIZATION zTr.~O~ ~ B 4000 27.6

2ol-- , r , .~ 'Cr , C 4600 31.0 D 500O 3 4 . 5

n A I ! "o 5 lo Is 20 zs 3o 35 40

NORMAL STRESS (MPa)

Fig. 3 . M o l a r diagram for reservoir chalk illustrating the effect of increasing pore pressure for cases where K = 0 . 2 a n d K =

0 . 5 .

There is also evidence from field observations [cores, FMS (Formation Microscanner, a micro-resistivity tool for observing borehole walls) logs, and FSMT (Formation Subsidence Monitoring Tool, a logging tool for quantitatively determining the compaction that occurred during a period between logging runs) measurements] that shear-induced fracturing likely has

11% ROCK MECHANICS IN THE 1990s

occurred in the reservoir. Cores recovered from recently drilled wells show evidence of massive fracturing, or fragmentation, and are consistent with FMS logs in the same wells. In addition, FMST measurements in injection wells have yielded results that can be interpreted by assuming that pressure increases cause additional compaction of the affected chalk. The observations of increased compaction of fragmented reservoir rock support an assumption that has been made in modifying the DYNAFLOW subsidence model to account for shear- induced compaction. The modifications to the DYNAFLOW subsidence model will be discussed later in the paper.

Mechanical Behavior of the Overburden Material The mechanical properties of the overburden play an

important role in controlling both the amount of compaction that occurs in the reservoir and just how much of the compaction that does occur is transferred to the seafloor as subsidence. As the reservoir starts to compact due to a reduction in pore pressure, and typically the compaction will occur dominantly in the central portion of the reservoir where porosities are highest, the overburden will move downward but will tend to resist the deformation; i.e., it will tend to remain as a rigid structure over the reservoir, flexing downward over the crest of the reservoir, but remaining more or less pinned in place over and exterior to the flanks of the reservoir. As this occurs, a portion of the load that previously was supported by the reservoir rock in the center of the field is transferred to the flank regions of the reservoir. The phenomenon involved here has been referred to as the "arch effect."

The effectiveness of the arch in resisting deformation depends on the mechanical properties of the overburden material, essentially on its rigidity characteristics or its ability to support a shear stress. As the process evolves, the overburden is flexed in such a way that shear stresses develop and have maximum values nominally over the flanks of the reservoir where there is a rather distinct transition from high-porosity chalk to low-porosity chalk, i.e., where bending of the overburden is greatest. If the shear stresses become sufficiently large, the material loses its ability to resist deformation. When this happens, the effectiveness of the arch is diminished, and the load- transfer feature becomes less effective.

In the DYNAFLOW modelling work presented here, the overburden has been described using the Drucker- Prager [6] procedure for treating the yielding process in solids. Parameter values for the overburden were developed in prior studies [5] and were not varied during the present study.

SHEAR-COMPACTION MODEL FOR RESERVOIR CHALK

reservoir, and because a logical argument can be made that an assemblage of fractured or fragmented chalk (either naturally fractured rock or rock that has been fractured by shear stresses during production or injection operations) will tend to compact more (i.e., appear weaker) than intact chalk, it was concluded that shear- induced compaction is a likely candidate for the mechanism (or possibly one of the mechanisms) that has been missing from previous simulation models, This is not to say that shear failure itself leads directly to compaction, but rather that the evolving stress state in a segment of the reservoir makes it possible for matrix blocks or fragments of chalk to be slightly displaced relative to each other, thereby changing the nature of localized stress distributions within the fragments so that nonuniform compaction of individual fragments can occur, probably by pore-collapse compaction. To clarify, in a designated volume within the reservoir that contains a very large number of fragments, some of which may be newly formed, each fragment will tend to move slightly relative to neighboring fragments (by translation and/or rotation) when shear and normal forces across the irregular surfaces separating the fragments permit. This slight movement will lead to the development of localized zones of concentrated stress where compaction can occur by pore-collapse. This is illustrated on Figure 4. As a result of the localized compaction, the shapes of the fragments are changed, thus creating a situation in which the collection of fragments can be packed more effi-

Sites of High Stress/ Large Compression

Adjacent Fragments Rotate Relative to Each Other

c .o_

c~ e-

o

o ¢: ¢L

E O ~J

¢l E

Q.

Sites of High Stress/ Large Compression

Adjacent Irregularly Shaped Fragments Translate Relative

to Each Other

Because observations from the field support a conclusion that shear failure has occurred in the

Fig. 4. Illustration of how shear-induced displacements create sites of high compressive stress in rock.

ROCK MECHANICS IN THE 1990s 1197

ciently. The net result is that the zone of rock containing the fragments will undergo increased compaction, or in effect that the affected rock will appear to be weaker.

Because of geometrical and other physical constraints (e.g., the thickness/width ratio of the reservoir is very small and the outer boundary, because chalks in the flanks tend to be of low porosities and high strength, is not displaced significantly), the overall, or global, deformation of the reservoir must occur under conditions of uniaxial strain. Uniaxial strain conditions also must be maintained on a more local scale (i.e., for zones of reservoir rock having lateral dimensions comparable to thicknesses of compacting intervals) as well because, on this scale, adjacent zones of compacting rock push against each other and by so doing prevent significant lateral displacements. These constraints on displacements, which exist in the reservoir but not in typical uniaxial- stain compression tests in the laboratory, are probably also responsible for the K value in the field being small compared with K values from laboratory uniaxial-strain compaction tests. The small displacements of fragments relative to neighboring fragments that occur when in-situ stress conditions permit, which is a concept that lies at the heart of the shear-induced compaction model, must occur within the general and overriding constraint of uniaxial strain. The line of logic discussed above has been pursued in developing a constitutive model for reservoir chalk that permits enhanced compaction to be realized as a consequence of shear.

A key feature of the model is that the stress path followed during depletion of the reservoir (pressure decline) is such that the ratio of the change in horizontal effective stress to the change in vertical effective stress (K) is 0.2 (actually, the value can be specified). This value for K leads, as expected, to the development of large deviatoric stresses in the reservoir, which eventually lead to a condition where a critical envelope for shear- induced compaction is encountered. The critical envelope in the model is actually a conical surface in 3-D effective stress space that is symmetric around the hydrostatic stress line, but for ease in explanation it will be described here using a Mohr diagram (shear stress versus normal stress). This is depicted on the upper portion of Figure 5, where key parameters of the model are defined. The existing stress state on this diagram is represented by a Mohr circle, and its position relative to the critical envelope is identified by the "mobilized friction angle." The weakening feature of the model is illustrated on the stress-strain graph in the lower portion of the figure.

In developing the model, a convention of referring to the envelope where conditions for shear-induced compac- tion are realized as the "critical envelope" has been adopted to avoid confusing this envelope with other failure envelopes that are commonly used to identify conditions where a computational code is instructed to perform a numerical calculation using an established failure procedure based on (for example) the Von Mises or Drueker-Prager criterion. An "ultimate failure enve-

UFE

C: " ~ / * " ~ , ~ ~ ' v ~_ ..... ,ohr Circle ': ~ / . . . . . . - j ~ [ ~ Representing / : \ *ie ngSmS

" ~ ¢ , e - 0 - - EFFECTIVE NORMAL STRESS

s i n ~,,, =

Mobilized Friction Angle. Its value indicates position of existing stress state relative to CE and UFE.

[ ~ - o ' , , , , ] / [ z~ + (o~ + ~ ) ]

SS o SS,

~ by l/f Chalk Immedietaly after Failure

I ~ SS 0 = Stress-Strain Curve Showing

Pore-Collapse Compaction for Intact Chalk

I SSu = Stress-Strain Curve for Chalk when

Fully Weakened by Shear Failure

VERTICAL STRAIN

Fig. 5. Molar diagram and stress-strain graph illustrating salient features of the shear-induced compaction model for reservoir chalk.

lope" is also defined in the shear-induced compaction model, and could be used to trigger some type of failure or yielding calculation, but in the present model is not. Instead, this envelope is simply used as an aid in defining the input parameters and for other features of the model that are in place but to date have not been exercised.

The critical envelope, the ultimate failure envelope, and the line drawn tangent to the existing-state Molar circle are all anchored at the point A (called the "attraction") shown on Figure 5. By monitoring the value of the mobilized friction angle, it is possible to determine where the existing stress state is relative to the critical envelope. Input quantities required to specify the critical and failure envelopes are the friction angle and cohesion for the ultimate failure envelope (which are used internally within the code to determine the attraction) and the friction angle for the critical envelope.

When the Mohr circle has grown to the point that it intersects the critical angle, the code is instructed to change from one stress-strain curve for the material in the affected finite element cell to a weaker curve, as is

1198 ROCK MECHANICS IN THE 1990s

depicted on the lower portion of Figure 5. The compaction behavior prior to encountering the critical envelope is described by the normal pore-collapse compaction curves for chalk that have been used in the past (Figure I), and afterwards by the weaker stress- strain curve. Once the transition to the weaker curve has taken place, return to the stronger one is not permitted. The weakened stress-strain curve is taken to be a multiple of the original pore-collapse compaction curve, and the multiplieative factor relating the two stress-strain curves is called the weakening factor (WF). There is no strong reason for representing the weakened stress-strain curve in this way (i.e., as a multiple of the original pore- collapse compaction stress-strain curve) except for the notion expressed previously that the localized compaction within the fragments which leads to the enhanced packing efficiency likely occurs by pore-collapse compaction.

The stress path followed in the model is controlled by the K value from the field observations. During application of the gravitational load while initiating a computational run, the K value is set at 0.5 so that stress conditions believed to exist in the reservoir at discovery [nominally 9000 psi (62 MPa) total vertical stress, 8000 psi (55 MPa) total lateral stress, and 7000 psi (48 MPa) pore pressure, which translate into 2000 psi (14 MPa) effective vertical stress and 1000 psi (7 MPa) effective horizontal stress] will be realized. Once initial conditions have been established, the value for K is changed to 0.2, and that value is held as long as the vertical strain is compressive, which normally corresponds to a condition of declining pressure or increasing effective vertical stress. When the vertical strain direction is reversed, indicating extension (in most cases triggered by increasing pore pressure) in a cell, the K value is set to 0.5; this value is maintained until there is another change in strain direction. The K value that ought to be used during extension, or repressurization, is actually not known, and could be as large as 1.0. A value of 1.0 was assumed for the illustrations on Figure 3. For applications of the model considered in the present paper, which involve use of average pore pressures in relatively large cells of the model, pressure increase magnitudes are rather modest [less than about 100 psi (0.7 MPa)], the weakening feature of the model during repressurization to be of little consequence. The shear-induced compaction for the cases of interest here has been brought about by pressure decline. In cases involving the explicit treatment of rock response near injections wells, where pressures increase magnitudes may be several thousand pounds per square inch (tens of megapascals), it is important to correctly specify the K-value.

APPLICATION OF SHEAR-INDUCED COMPACTION MODEL TO EKOFISK

General Information In the process of implementing the shear-failure model

in the DYNAFL£)W reservoir compaction/subsidence

model, more than forty simulation runs have been performed. These runs were made for a variety of reasons, notably: (1) to learn as much as possible about the physical processes that are being modelled; (2) to determine the degree to which assumptions of the model are realistic, (3) to determine the sensitivity of computed results to variations in the values of input parameters, (4) to develop a set of parameter for the reservoir rock and overburden that will yield good agreement between computed results and measurements that have been made in the field, and, ultimately, (5) to provide a valid means to predict the future course of reservoir compaction and subsidence for any reservoir management scenario for Ekofisk.

In this paper the results of two simulation runs for the Ekofisk field are presented to demonstrate that the model performs as expected and provides results that are in good agreement with field measured data. The two most important features of the shear-induced compaction model that are controlled by values of input parameters are the positions of the critical envelopes (CE on Mohr diagram of Figure 5) in effective stress space and the multiplier (the weakening factor, WF) used to define the weakened stress-strain curve (SSu) relative to the initial pore collapse stress-strain curve (SS o) for each of the chalk materials in the model (representing different porosities and different quartz concentrations). The timing of the transition to the weakened stress,strain curves is controlled by the positions of the critical envelopes, while the actual effect of the transition is controlled by the magnitude of the weakening factor. The simulation runs selected for presentation here illustrate, among other things, that the model performs as expected and gives good agreement with key field observations.

Comparisons between Calculated and Measured Subsidence

Each run of the 3-D, full-field compaction/subisdence model for Ekofisk provides massive quantities of data pertinent to displacements and stresses throughout the entire volume of the reservoir and surrounding medium considered in the model. In this paper, attention is given only to vertical displacements (subsidence) at a point on the seafloor in the central portion of the field where the subsidence bowl is nominally the deepest. Figure 6 and 7 show comparisons of calculated and measured subsidence and subsidence rate for two model runs. The runs represented here were made using parameters for the critical envelopes that were derived from iabortory experiments [9]. The two curves on Figure 6 illustrate how calculated subsidence values depend on realistic variations in magnitudes for the cohesion, the intercept of the critical envelope with the shear stress axis (see Figure 5). While not necessarily evident from the graph, the runs show that shear-induced compaction starts with the high-porosity (38-40%) chalks in the late 1970's and early 1980's and continues with the mid.porosity 04- 36%) chalks during the mid- to late 1980's. For the

ROCK MECHANICS IN THE 1990s 1199

8

7

E 6

LJ 5 (_~ z 4 LIJ

@3 GO []3 D2 (/3

I

0 - 1970

Run #37

• Measured data

loses of Calculations: Pre 1992 - Field History Post 1992 - Premised Future

t I t I I I I v

1980 1990 2000 2010 TIME (year)

Fig. 6. Measured subsidence from the Ekofisk Field compared with calculations based on the shear-induced compaction model. Parameters used in the model are discussed in the text.

particular pressure-time input data file used for these runs, the lower porosity (30-32%) chalks experienced some shear-induced compaction, but not as pervasively as the higher porosity materials. The weakening factor used for all chalk materials in both runs is 1.5. As is evident, the two computer curves bracket the measurements quite nicely.

80 "C" >, 70

E 60 U

5O Ld I - - < 40 r~

30 L.d ~z 20 c~ 10 03 []3 0

m- lO 197~

- - Ave'rage subsidence 'rate cleduced from the field measurements

_r-L. Computed resu l t s (Runs 37 end 40 Averaged)

I I I I l I I I

1980 1990 2000 2010 TIME (year)

Fig. 7. Measured subsidence rates from the Ekofisk Field compared with calculated rates based on the shear-induced compaction model. The calculated rates are averages for the two runs depicted on Figure 6.

Measured and calculated subsidence rates deduced from the information on Figure 6 are shown on Figure 7. The subsidence data (data points) on Figure 6 for times from 1985 and beyond (i.e., after the discovery of subsidence at Ekofisk) are based on actual measurements of subsidence (bathymetric surveys, satellite surveys, and seafloor pressure gauges), while the earlier data are more properly described as estimates that are based on photographs and other historic records. The data on the graph for the period after 1985 represent only a small fraction of the subsidence measurements that have been made and correspond to subsidence values at times

between periods during which subsidence rates have been observed to be substantially constant. The nominally constant subsidence rates during these periods are shown as the heavy horizonal lines on Figure 7, and were determine from the slopes of straight lines that connect the points on Figure 6. Calculated subsidence rates shown on Figure 7 represent average rates for the two simulation runs; these average rates are presented in a manner to facilitates comparison with the measurements. The agreement here is also considered to be good.

The subsidence and subsidence rate data on the figures for times beyond 1992 do not necessarily represent valid predictions because the actual operational (production and injection) plan pursued in the field may not be consistent with that premised for the compaction/subsidence simulation model runs discussed in this paper.

DISCUSSION OF SHEAR-INDUCED COMPACTION AND CONCLUSIONS

The shear-induced compaction model described in this paper was formulated on the basis of several significant observations from the field and laboratory: (1) deviatoric stresses in the reservoir have become very large during the production life of the Ekofisk field, (2) laboratory tests to examine failure processes in chalk revealed that shear-induced fracturing and/or slippage on existing or newly formed fractures can readily occur under the stress conditions that have evolved in the reservoir, and (3) recovered core and observed conditions in boreholes indicated new fractures likely have been formed and that slippage on fractures has also occurred. As these observations were being made and their interpretation was evolving, it was also becoming quite evident that subsidence rates in the field were tending to be larger than subsidence rates calculated with existing numerical simulation models that accounted only for pore-collapse compaction of the chalk reservoir rock. The possible link between the various observations led to the conjecture that there may be an additional mechanism for compaction in the reservoir that was related to the development of large shear stresses. As a result, a constitutive model was formulated to treat shear-induced compaction and incorporated into the full-field, 3-D compaction and subsidence model for Ekofisk. Results of calculations based on the shear-induced compaction model presented in this paper show that the discrepancy between measured and calculated subsidence and subsidence rates can readily be eliminated.

At this time we are pleased with the agreement between computed and measured displacements in the central portion of the field. Work is continuing to examine other aspects of the model and of the compaction and subsidence phenomena, e.g., displacements elsewhere in the modelled regime, changes in stress, relationship between compaction volumes and subsidence bowl volumes, and how properties of the overburden influence compaction and subsidence.

1200 ROCK MECHANICS IN THE 1990s

Subsequent work likely will lead to refinements in model parameters, particularly to improved definitions of stress- strain curves for chalks that have experienced shear- indueed compaction. A minimal refinement is expected to be the specifying of a unique value for the weakening factor, WF, for each material, with values closer to unity (than the 1.5 value deduced from the present work) being used for high-porosity chalks (because of the large pore- collapse compaction already experienced by these materials) and perhaps larger values for low-porosity chalks. In the meanwhile, the primary objective of developing a model to bring measured and calculated subsidence magnitudes and rates into alignment has been met, as is evidenced by the good agreement between measured and calculated subsidence magnitudes using realistic values for the parameters of the shear-induced compaction model.

Acknowledgments - The authors acknowledge with thanks permission to publish this paper from Phillips Petroleum Norway and coventurers, including Fina Exploration Norway u,a.s., Norsk Agip A/S, Elf Petroleum Norge AS, Norsk Hydro a.s., Den norsk stats oljeselskap a.s., Total Norge A.S., Elf Rex Norge A/S and Norminol A/S. The opinions expressed in the paper are those of the authors and do not necessarily represent those of the Phillips Norway Group.

REFERENCES

1. Boade, R. R. and Chin, L. Y., The DYNAFLOW Proce- dure for Simulating Ekofisk Subsidence with Results for Two Spring 1986 Reservoir Management Scenarios, Phillips

Petroleum Company, Production Technology Branch, Research and Services Division, Research and Develop- ment, Bartlesville, Oklahoma, November 1986.

2. Boade, R. R., Chin, L. Y., and Siemers, W. T., Fore- casting of Ekofisk Reservoir Compaction and Subsidence by Numerical Simulation, OTC 5622 presented at 20th Annual Offshore Technology Conference, Houston, Texas, May 2- 5, 1988.

3. Boade, R. R., Chin, L. Y., and Siemers, W. T., Forecasting of Ekof~k Reservoir Compaction and Subsidence by Numerical Simulation, JPT (July 1989), pp 723-728.

4. Chin, L. Y. and Boade, R. R., FuU-Field, Three- Dimensional Computer Modeling of Reservoir Compaction and Subsidence at Ekofisk, Phillips Petroleum Company, Production Technology Branch, Drilling and Production Division, Exploration and Production, Bartlesville, Oklahoma, April 1991.

5. Chin, L. Y. and Boade, R. R., Full-Field, 3-19 Finite- Element Subsidence Model for Ekofisk, Third North Sea Chalk Symposium, Copenhagen, June 11-12, 1990.

6. Prevost J. H. DYNAFLOW Manual, Princeton University, Princeton, New Jersey (1983).

7. Sulak R. M., Thomas, L. K., and Boade, R. R., 3-l) Reservoir Simulation of Ekof~k Compaction Drive, JPT (October 1991)pp 1272-1278.

8. Teufel, L. W., Rhett, D. W., and Farrell, H. E., Effect of Reservoir Depletion and Pore Pressure Drawdown on In Situ Stress and Deformation in the Ekofisk FieM, North Sea, 32nd U.S. Symposium on Rock Mechanics, University of Oklahoma, July 10-12, 1991.

9. Rhett, D. W. and Teufel, L. W., Failure Criteria for High Porosi~ North Sea Chalks, Fourth North Sea Chalk Symposium, Deauville, France, September 21-23, 1992.