Reservoir Compaction and Seafloor Subsidence at Valhall

.

-.

SPWISRM 47274

Reservoir Compaction and Seafloor Subsidence at ValhallP. D. Patti[lo, SPE, Amoco Production Company, and T. G. Kristiansen, SPE, Amoco Norway Oil Company, and G. V.Sund, Amooo Norway Oil Company, and R. M. Kjelstadli, SPE, Amoco Norway Oil Company

AbstractThe combination of in situ effective stress and reservoir chalkmechanical weakness in the Valhall field has resulted incompaction and associated subsidence at the mudline. Thispaper summarizes recent fiite element modeling of thisphenomenon, emphasizing items of concern in a numericalsinmlation of this nature. The paper also presents new resultsfrom a simulation allowing inelastic overburden response.

IntroductionThe Valhall field is an initiaIIy overpressured, undersaturatedUpper Cretaceus chak reservoir located in the central grabenin the Norwegian sector of the North Sea. The reservoir is at adepth of approximately 2400 m subsea and consists of two oilhearing fmznatiom the Tor and Hod, The former containsroughly two-thirds of the oil and is a soft chalk characterizedby high purity (95-98% caIcite), high porosity (up to 50%) andhigh oil saturations (90% and greater)’. Oil and gasproduction from the field began in October, 1982.

The Tor discovery pressure was ody 3.4 MPa less than the48.3 MPa overburden S-SS, implying minor formationcompaction during burial. With depletion, the Tor inparticular has compacted significantly, inducing tubularfailures in the ~servoir and overburden, and subsidence at thenmdline. This paper ignores @buIar design issues and focuseson ~dicting Valhall mudline subsidence.

Subsidence p=diction is not new to the petroleumindus&, or even to the central graben of the North Seal’3”7.The model developed for Valhall, however, does contain_ features deserving review. These features, and their~e in subsidence modeling, are detailed in the

discussion to follow.

Previous WorkStudies of varying complexity have attempted to predictreservoir compaction and associated mudline subsidence in theValhall field. Previous finite element studies include an earlyanalysis performed by the Sediment Deformation ResearchCenter of University College, London for the NorwegianPetroleum Directorate’ and unpublished analyses performedboth by engineering consultants and Valhall partners. Inaddition to numerical models, Ruddy et aI.* generated mudlinesubsidence predictions using an elastic su~rposition modelthat has reservoir compaction as an input variable. Generalconclusions reached from these studies include the following1.

2.

3.

4.

All models illustrate the arch that normally forms over thecompacting region.bmparison of twodimensional and three-dimensionalmodels of Valhall indicate that, for this reservoirgeometry, two-dimensional (plane strain) modelsoverpredict overburden flexibility and, thus, mud linesubsidence.A shortcoming of simpler models based on elasticitysolutions to half-space geometries is the lack of detailprovided on the near reservoir stress state. This omissionbecomes critical if well failures in the vicinity of thereservoir/overburden interface necessitate a closerexamination of this region.The coarser meshes usually necessarv to accommodate thesize of reservoir subsidence models make predictionssensitive to in~rpolation schemes used to assign porepssures and rock mechanical behavior.

Reservoir Model InterfaceThe driving force for compaction in the reservoir is effectivestress increase associated with pore pressure depletion.Spatial and temporal pore pressure distributions used in thisstudy were retrieved directly from executions of an in houseresrvoir model. In particular, elements of reservoir simulatorinputioutput data used by the finite element model include thefouowing1. Spatial positions of the model grid points.2. The porosity at each grid point.3. Initial and subsequent pore pressure distributions used,

3n

2 P. D. PAITiLLO, T. G. KRISTIANSEN, G. V. SUND, R. M. KJELSTADLI SPE 47274

respehly, to set the initial effective stress distributionin the reservoir and subsequent changes to the initialdistribution.

AIthough the ftite element model and the reservoir modelshare common grid points, the meshes are not identical.Carefi interpolation and extra@Iation procedures were usedto recover data from the reservoir medel for use in the ftiteelement amdysis.

No attempt was made to model the coupling between- mechanical deformation of the reservoir and surrounding rock,

as predicted by the ffite eIement model, and flow in thereservoir. This is not to say that the important driving forceaswbd with reservoir compaction is absent from the fluidflow analysis. The reservoir sirmdator does take account ofthe effect of reservoir rock compress]%ility on fluid flow.However, in the reservoir simulator, each computational cellacts in isolatioxL That is, there is no accounting in thereservoir model for bridging effects associated with interactionbetween relatively strong and weak regions of the reservoir.To fi.dly model such an effect given the currently availabletools, it wotid be necessary to either resort to a fully coupledporn-mechanical numericaI forrmdation or intersperse eachtime step m the reservoir sirrndation with a finite element loadstep, using the results of each model (pore pressure from thereservoir @tor, pore volume strain from the finite elementmodeI) as input in the subsequent step of the other.

~ impact of resewoir fIow/kinematic coupling onVaIhrdl mm is difficult to estimate. One would expectbridging, in particular, to influence reservoir pore volumechange, with the resulting altered pore pressure distributioninfluencing subsequent mechanical deformation. Any changein the reservoir respu will also affect the values ofparameters, such as rock compressibility and reservoirproperty distributions that have been fine tuned during the lifeof reservoir sinndation at Valhall. me situation is furthercomplicated by the fact that, at present, the finite element gridis not as fine as the reservoir simulator grid, such thatimpo~t l~d effects Of property variation in the reservc)ir

and near reservoh region are not fully detailed. AII of this isnot to say that coupling should not be the ultimate goal inmodeling stress sensitive reservoirs such as ValhalI. Oneshodd expect, however, that a significant period ofreadjustment will accompany each step toward an integratedporo-mechanical modeI.

Finite Element ModelThe ftite element model constructed in this study Figs. 1 and2) employs the ABAQUS general purpose finite elementprogram.* Element C3D20RP, a 20 node, reduced integration,quadratic displacement, linear pore pressure element was usedfor all reservoir and non-reservoir strata. Underburden andsideburden boundaries were modeled with elementCIN3D12R, a 12 node, reduced integration, quadraticdisplacement semi-tilnite element that does not have porepressure capabilities.

All depths in the model were referenced to the rotary kellybushing (RKB).

Nonreservoir Eiements. Four overburden layers appear inthe model representing, from the mudline to the reservoir, thePliocene (including the Pleistocene and Miocene), LowerMiocene, MiddIe Eocene, and Paleocene, with the top of thePaleocene coincident with the Balder formation. ThePaleocene was taken to have a constant thickness of 40 meters.All other overburden layers have constant depths-to-top asgiven in Table 1.

E&astic Behovwr. In this study, elastic stiffnesses aretaken from a petrophysics analysis using the following sources:1. Logs taken on a wellbore which does not penetrate the gas

cloud overlying the Valhall reservoir were processed withshale strength correlations.

2. A search of an industry data base yielded ratios of normaland in-plane sonic velocities, which were used to ratio in-plane properties to those associated with the normaI.

The results of this work are summnrized in Table 2. Eachoverburden layer modeled is assumed to exhibit planarisotropy, with greater stiffness in the horizontal plane.

Sideburden elements (outside the reservoir simulator grid)were assumed to be elastic wifi elastic properties identical tocoincident elements in the reservoir/overburden cohmm. Theunderburden and sideburden elements do not contain porefluids.

Plastic Behvior. This study is the fust application ofinelastic behavior to the Valhall overburden. The ABAQUScap plasticity model (see Appendix) was used to modeIplasticity in the overburden, as well as modeling the chalkreservoir.

Fjaer et aI? report Mohr-Coulomb parameters whichprovide reasonable guidelines for defining the Drucker-Pragerfailure surface. Specific values for these and other cap modeIparameters selected for the Valhall overburden are given inTable 3. In all instances except the Lower Miocene,a‘ = 0.01, R = 0.5. For the lower Miocene, R = 0.35 toremove an inconsistency in the data fitting process. Thisadjustment is far removed from the point where the initialyield surface is penetrated and should, therefore, have minimaleffect on the numerical solution.

Fig. 3 illustrates the work hardening curves used to followmovement of the cap. The procedure used to constnict thesecurves ignores tie transversely isotropic nature of the non-reservoir material. This additioml complexity was notincluded in the development of Fig. 3, a simplification whichmay be justified by the limited knowledge of the overburdenand underburden material.

Reservoir Elements. The ftite element mesh in the reservoiroverlays the reservoir model grid and is approximately one-sixteenth as dense, the lateral extent of each reservoir Iayerbeing 17 by 9 elements, with six reservoir layers modeling twolayers in the Tor formation and four layers in the Hod

378

SPE 47274 RESERVOIR COMPACTION AND SEAFLOOR SUBSIDENCE AT VALHALL 3

formation. Increased density within the lateral boundaries ofthe reservoir is desirable as it has the potential for renderingthe model more flexible in two areas:1.

2.

Definition of ~ properties. The primary factor indeterrninin g variation of chalk strength is porosity. Afiner mesh permits a more detailed deftition of porosityvariation in the reservoir, and, therefore, allows someeIements (tiose with higher porosities) to undergocompaction due to pore collapse at an earlier stage in theload history,Even with the detail used in the present model, a numberof elements in the reservoir have large lateral extents ascompared to vertical extent. Such high aspect ratios canmsr,dt in artificial, numerical stiffening of the mesh inresponse to shear stresses in the vertical plane.

Materird properties in the reservoir were varied on anelement by element basis, the parameter fixing exact materialc~ being the average porosity within an element,

Material behavior for aII chalk elements was taken frominternally deveIoped correlations, where elastic behavior isassumed to be isotropic, and inelastic behavior is describedusing the ABAQUS cap plasticity model. The primaryvariable in correlating both elastic and inelastic parameters isporosity. Fig. 4 indicates the accuracy of tire ABAQUS model~ +ucing Vamrdl type curves by Andersenlo.

Boundary Conditions. The following boundary conditionswere ~Iied to the model:1.

2.

3.

ti- lowermost Iayer of nodes in the serni-itiiniteunderburden and nodes on lateral surfaces of the model(i.e., the serni-infiite elements representing theaidehurdcn) were freed.Pore pressures in the reservoir proper were taken from the~oir simulator.Pore pressures in the overburden were held constant.Possfile undrained response was not considered.

Initial Conditions. The overburden is assumed to be in aninitird (discovery) state of incipient plasticity on the cap. Thisconclusion is based on an analysis of overburden stress-straincurves used to estimate the pre-consolidation stress. Assumingmdaxird strain compaction of the overburden material, one canuse the maximum effective overburden stress from the pre-consofidation stress estimate to back calculate a pore pressurefrom the =ve _ principle. Since this pore pressureestimate correlated weIl with the current pore pressure profile,it was concluded that the maximum effective vertical stress thematerial has experienced is equal to the current, and thereforetie material is located on the cap”. Defining (tensionpositive)

{-}p=-~fr u , ............................................................... (1)

iq=~= 2.’:0’

2--. ................................................. (2)

where J2 is the second invariant of the deviatoric effective

stress tensor, m’, and

0’ = u+ p I .................................................................... (3)---

consider a biaxial stress state where the two nonzerocomponents are the vertical stress, Uv , and the isotropic

horizontal stress, ah. For this simpIe stress state,

q=o~-erv ...................................................................(4)

p=-;(ov +2crh) ....................................................... (5)

If it be further assumed that the initial state is one ofuniaxial deposition, then for a transversely isotropic materialwith hotintal bedding,

Eh Vhv

‘h =~ 1– vh 0“ “"""""""""""o""""""o""`"""""""`""""""""""'""""""""""""'(6)

Substihrting this expression into Eqs. (4) and (5), wearrive at the useful relations

q=(’-?%)ov ~................................................ (7)

(p.;1+2%‘hv )—uEvl-vh

v ) ......................................... (8)

,_q v~v

E, l-vh Ev(l-vh)-Ehvhvq=s

,+2~ v~vp.3

EV(l – vh )+ 2Ehvhvp. (9)

Ev l–vh

me last relation gives the load history in p – q space and

allows one to compute the intersection of the depositionhistory with the yield surface, thus determining the stress statecorresponding to incipient yield. Table 4 summarizes theintersections of Eq. (9) with the ABAQUS yield surface cap(Eq. (A-1)), given the material properties from Tables 2 and 3and Fig. 3.

The slope of the load path, taken from Table 2 and Eq. (9),is in every case less than ~, substantiating the cap, rather than

the Drucker-Prager failure surface, as the portion of the yieldsurface penetrated. Note that the load path is sensitive toPoisson’s ratio, which appears to be very high for aIIformations except the underburden. T&ing the Paleocene asan example, any value of Poisson’s ratio less than 0.33 wiIlretit in a load path slope greater than the Drucker-Pragerfailure surface and thus, depending on the value of d, increase

379

* ,.

u-%

4 P. D. PATTILLO, T. G. KRISTIANSEN, G. V. SUND, R. M. KJELSTADLI SPE 47274

the Likelihood of penetrating this surface, rather than the cap.Pore pressure at the mud line (top of Pliocene/Miocene)

was set at 0.6913 MPa. The interface between thePlioc ~ene and Lower Miocene was assigned a porepressure of 1522 kg/m9. The interface between the bwerMiocene and MlddIe Eocene, was assigned a pore pressure of1618 kg/m3, The interface between the Middle Eocene andPaleocene was assigned a pore pressure of 1642 kg/m3,

The pore pressure at the bottom of the Paleocene was givenspechd treatment. A conflict arises at the Paleocene/reservoirintefice between the relatively smooth pore pressure variationassigmnen~ above and the more erratic values in the Tor taken* the reservoir simulator, Further, abrupt variation in pore~ssure cmi tiect both the initial stress and subsequent stresspaths. To minimize this effect, a double layer of coincidentnodes was created at the PaIeocendreservoir interface, withone Iayer assigned to Paleocene elements and one layerassigned to reservoir elements. Coincident nodes wereassigned differing pom pressures, according to whichformation they were attached, but were constrained to haveidentical displacements.

Reservoir Elements. Reservoir elements were assigned aneffective stress composed of a gravitational body force due toa total specflc weight of 0.02 I MN/m3 and a pore pressuretaken born the reservoir simulator, An inconsistency dmsexist between the defiition of material properties in thereservoir and the pore pressures used to initialize the model.The total specific weight is assumed constant, and for tiepm’POSCof CaICUlath3gthe gravity load the pore fluid isassumed to have a specific weight of 0.00999 ~/m3. Thusthe dry rock specific weight is adjusted for each reservoirelemen~ depending on its porosity, to yield the constant 0,021MN/m3 gravitational Ioadii.

me effective horizontal to vertical stress ratios were takenas cons~~, with of /o~ = 0.735 in the Tor layers,

cr~/u~ = 0.508 in the upper two Hod layers and

Oi /oJ = 0.577 in the lower WO Hod layers.

Bou- and Unde&utien (Semi-i@nite) Elements.Elements not having pore pressure capability must beini- with total stresses to equilibrate them with the porepressure eIements. If the reIation between total stress andeffective stress is taken as

51=0/ – au ................................................................(lo)

for the ith component of normal stress, and where u is pore~ssure, a = 1- K/K= , where K is the buk modulus of the

porous matrix and Ks is the buk modulus of the solid,

–sv-o-2%lau*ooooo(11)‘h ‘hv‘h=~l-vh

for transvemely isotropic formations.Fortunately, the additional pressure terms do not require

the total swss ratio to vary with depth, provided the ratio of

pore pressure to total vertical stress is constant. For theoverburden, this requirement coincides with the gradientassumptions listed above (The parameter ez was assumed tobe unity for all layers for the purpose of assigning initialstresses.) For the sideburden elements opposite the reservoir,the situation is more complex, as the pore pressure gradient isnot constant. As a compromise, the reservoir sideburden andunderburden were assigned an initial total stress ratio identicalto that of the Paleocene immediately above. Inconsistenciesbetween this simpWled initial s@ess profile and that requiredto bring the entire model to equilibrium were left to resolutionin the fiist load step.

Mutine Elements. The elements adjacent to the mudlinewere loaded by a vertical stress of 0.6913 MPG representingthe hydrostatic load of 69.2 meters of sea water (0.00999MN/m3).

Numerical Procedure. By far the most difficult portion of a-reservoir subsidence analysis is achieving initial equilibrium ofthe overburdedreservoirlunderburden under the stressdistribution assumed to exist at the onset of the calculation.This &lculty arises from the fact that all contributors to theinitial stress state are not known to a stilcient degree ofaccuracy. As long as the substrata are horizontal and the porepressures vary linearly, even inaccurate values for theformation densities and in sim pore pressures will yield anequilibrated initial state rather quickly. Unfortunately, formost problems, the substrata, particularly the reservoir, are notstrictly horizontal. Furthermore, the pore pressure distributionretrieved from the reservoir simulator is variable throughoutthe reservoir. In such cases, it is often not possible to set fortha consistent initial stress state from which to begin an analysis.

The procedure used to achieve initial equilibrium in thecurrent study assumes that one is relatively cotildent of hisknowledge of the ~tial stress state and/or is willing to acceptany inaccuracies embodied in the prmedure, The stepsrnvolved are as follows:1.

2.

3.

&ate a model input file as if no problem with the initialequilibrium state existed. This file will have ordy onestep, to achieve initial equilibrium. Before executing theamdysis, make the following changes to the model as partof the initialization process:a) Fix all nodes in the model in all displacement degrees

of fiedom. Include the necessary instructions towrite file output that wiIl save the nodaI reactionforces from this step.

b) Execute the analysis using the input as revised by theprevious step (I a).

Use the output horn the above execution (1 .b) and createa list of concentrated loads at all nodes in the mesh thatexactly counter the imbalance in equilibrium due toinaccuracies in the estimate of the initial stress state.Re-execute the analysis with the original input file editedto reflect the tru6 boundary conditions (i.e., remove theconstraints impowd in (1.a) and include the concentrated

380

—


loads from the previous arudysis (I b)). At the end of thesecomi executio~ the entire model should be equilibratedand ready to restart with pore pressure depletion steps todetexmine compaction and associated subsidence.

Note that with the procedure outlined above, a distributedload (representing the body force and vertical load in eachelement due to its weight and the weight of elements above it)is not rtecq ~. The initiaI s~ess state submitted to themodel should include the contribution of gravity, and thiscontribution is permanently incorporated in the model via theconcentrated loads calctited in the ~~st execution (1.b) andadded to the model as part of the second execution (3).

Time Steps. The major time steps were taken from theESSNOU Sh31Uhlti011. The entire depletion history wasconsidered to be a quasi-static process wherein transienteffects are ignored. Although each major time step wassubdivided into a number of increments, the sole purpose ofthis .suMvision was to achieve numerical convergence of thenoxdinear aspects of the model response to pore pressurevariation. Whhin a major time step, the pore pressure historyat each node was assumed to vary linearly between initial andfii vrdues for that step.

ResultsThe ftite element model consisted of 2613 elements with13541 nodes resulting in 35070 degrees of freedom. Theestimated RMS wave front for the model was 1468.

Platform Subsidence. Fig. 5 compares the model predictionof VrdhalI pIatform complex subsidence with fieIdme~ and with a previous, unpublished predictionempIoying an elastic overburden. Both models follow the fieldmeasuremerrt closely through the current history.

In a ti typical of these analyses, subsidence ispredicted to continue at a retively constant rate for the nextthree to four years, and then to proceed at a reducing rate ascompaction reaches its limits and the reservoir, viewedglobally, begins to work harden. This transition continuesthrough December 2031, the end of the simulation. Predictedplatform subsidence with the current model in December,2031, is 6.3 meters.

Fig. 6 dispIays the same data as Fig. 5 in rate form, wheretie measured data are presented as a twelve month movingaverage. Notice that the current study follows the ratemeasurements acceptably.

Two interesting conclusions cart be reached from Figs. 5and 6. First, note that the difference between the so-calledElastic OB curve and the Plastic C)B curve is minor. Thiswould imply that the overburden is not yieIding, that is, thatboth are actuaIIy elastic overburden curves. In support of thisconclusion, Fig. 7 illustrates the Ioad path taken by integrationpoints in the overburden immediately below the platform.Although the overburden elements are initialized at incipientyield, overburden response to reservoir compaction primarily

ties the form of unloading into elastic behavior. Only thePliocene/Miocene exhibits minor loading.

Secondly, it is worth noting that the elastic mechanica~properties for the Elastic OB and Plastic OB models are not ““”the same. The Elastic OB model uses earlier estimates ofYoung’s modulus and Poisson’s ratio (again assuming planarisotropy) that are softer than those used in the current analysis,presumably to partiaIly account for inelastic behavior.

It is tempting to attribute the close agreement between thetwo curves to the results of Berry and co-workers. Analyzingthe plane response at the boundary of a half-plane tocompaction represented by a displacement discontinuity, theyfound that the displacements are independent of the elasticconstants for an isotropic medium*2 and only dewnd on twodimensiodess combinations of the elastic constants for planarisotropy 13’lA.Here, however, the compaction is not specified apriori, but is part of the solution.

Figure 8 presents a comparison of compaction andsubsidence for the two models (Elastic OB and Plastic OB)along a plane passing roughly through the center of the meshalong its short axis. Note that the two models do not predictthe same compaction, and the close agreement on subsidenceoccurs primarily near the center of the mesh. It may be,therefore, that at least a portion of the cIose agreementbetween the two models is purely accidental.

Compaction and Mudline Displacements. Compaction andits associated subsidence is a direct consequence of effecti~cS*SS changes in tie reservoir associated with fluidwithdrawal. Initially, the reservoir rock matrix and resenoirpore pressure are equilibrated in their joint support of verticalstresses induced by the weight of the overburden. As fluid iswithdrawn, however, the reservoir pressure decreases, and anincreasing fraction of the overburden load is transferred to therock matrix. This increase in the effective stress on the rockmatrix is accompanied by deformation which is initially elasticin nature. For weak rocks such as the VaIhaIl chalk, however,continued pore pressure decreaseieffective stress increaseexc=ds the elastic limit of the reservoir matrix and theresponse becomes inelastic and accelerated.

The amount of compaction in a region of the reservoir isprimarily a function of the IocaI pore pressure decrease,porosity and thickness. Heterogeneity in the reservoir in aIItiese factors can lead to complicated compaction contours. Asan example, Fig. 9 summarizes the predicted displacement atthe top of the reservoir in December, 2031. The erratic natureof the contours is a direct result of the pressuredepletiotiporosity/thickness distribution in the reservoir.

In contrast, Fig. 10 illustrates tie result of transmitting thevertical displacements of Fig. 9 to the mudline. St. Venant’sprinciple acting through the intervening overburdensignificantly smoothes the character of the displacement field.This mitigation of displacement gradient is typical of analysesof the type’. .

A cursory examination of Figs. 9 and 10 might lead one to

361

9..,.

ti. !.

6 P. D. PAITILLO, T. G. KRISTIANSEN, G. V. SUND, R. M. WELSTADLI SPE 47274

conclude that compaction and subsidence are roughly equal.ActualIy, the gradients at tie top of the reservoir are-ienfiy steep to be missed by the contour interval selected(see Fig. 8). The predicted subsidence: compaction ratiocontinuously increases with time, reaching a vaIue ofapproximately 0.84 Iate in the simulation.

Horizontal displacements are also important at bothreservoir and mudline depths. In the vicinity of the reservoir,horizontal displacement gradients can damage tubulars. AtmudIine depth, horizontal displacement gradients can affectthe integrity of surface structures. Fig. 11 is a vector plot ofhorizontal displacements at the mudline. The generalhorizontal movement at the mudIine is toward the center of themxrvoir with a slight bias in the southeast direction (towardthe upper right of the figure), and with a maximum magnitudeof 2.45 meters in December, 2031. In the vicinity of theplatform, the honzontaI displacement in 2031 is approximately1.02 meters.

Closure Stress. Together with the material properties of thereservoir rmk and the assumed pore pressure history duringdepletion, the initial stress state, pardcularly in the reservoir,m~sents a major factor in determining the time history ofcompaction/subsidence. Given an accurate chaIk constitutivemodel and reservoir pore pressure, the closeness of the initialstress state to the yield surface can affect the point at whichpore volume coIlapse accelerates, and thus, the history ofmudline deformation. Field data collected during hydraulicfractming treatments are rhe best current means of determiningthe horizontal (closure) stress state within the reservoir.

As discussed above, initial horizontal stresses in thereservoir Iayers of the model m determined by the integratedoverburden, initiaI pore pressure and fixed (0.735 for Tor,0.508 for Upper Hod, 0.577 for tiwer Hod) horizontal tovertical effective stress ratios. These stress ratios weredetermined from an analysis of earIy history minifracs.

The current model predicts a trend of lowering closurestress with time. A similar, but less pronounced trend appearsin the field data. Comptig the two results, and ignoring acoupIe of anomalous points, it maybe generally concluded thatthe modeI is fairly accurate in predicting rninifrac closurestiss with time (on the order of 5% difference), slightlyove~dicting the field measurements.

Conclusions1. The current model sadsfactorily reproduces field

me~ of both mumdative and current incremental(rate) subsidence at the VaIhalI platform complex.

2. The ABAQUS cap plasticity model is an effective andaccurate means of’~ the constitutive behavior ofboth the overburden shaIe and the reservoir chalk.

3. Establishing the initial in siru effective stress state is anet and dificuIt part of compactiotisubsidencemodeling.

4. Inelastic behavior in the overburden in the form of

additioml compaction is not likely to be a major influenceon subsidence at Valhall.

Nomenclatured - q-intercept of Drucker-Prager failure surface,

m/Lt2, MPaE - Young’s modulus, tn/Lt2, MPa

&w - ~AQUS cap function, rn5t2, maG - shear modulus, rn/Lt2, MPaY2- second invariant of deviatoric stress, m2/L2t4,

MPa2k - ratio of yeild stress in triaxial tension to yield

stress in triaxial compressionK - bu~ modulus of porous matrix, m/Lt2, MPa

K, - bulk modulus of solid, rn/Lt2, MPaKt - see Appendix

@p= slope of Drucker-Prager failure surfacep - mean stress, m/Lt2, MPaP. - center of ABAQUS elliptical cap, fit2, Mpap, - intersection of ABAQUS cap with p-axis,rn/J.-t2,

IvfPa

q“m*~$*MpaR = measure on horizontal axis of ABAQUS capr - cube root of third stress invtiant, rn/Lt2, MPas - total stress, tit2, MPat - see Appendix, tit=, MPau - pore pressure, mfLt2, MPaa - Biot parametera’ - radius of ABAQUS transition circle~ - angle whose tangent is ~, degrees

#w, = volumetric plastic strainv = Poisson’s ratiocr - effective stress, flt2, MPa0’- deviatoric effective stress, m/Lt2, MPa

Subscriptsh -in the horizontrd planev - in the vertical plane

References1.

2.

3.

4.

5.

Ruddy, I. et al.: “Rock Compressibility, Compaction, andSubsidence in a High-Porosity Chalk Reservoir A Case Studyof VallraliField;’ JPT (JUIY 1989) 741.Geertsma, J.: “Land Subsidence Above Compacting Oil andGas Reservoirs,”JPT (June 1973)734.Mori~ N. et al.: “Quick Method to Determine Subsidence,Reservoir Compaction, and In-Situ Stress Induced by ReservoirDepletion,”JPT (Jan. 1989) 71.Chin, L Y., and Bosde, R. R.: ‘Tull-Pield, 3-D, Finite-ElementSubsidence Model for Ekofisk,” presented at the Third NorthSea Chalk Symposium, Copenhagen, June 11-12, 1990.Chii, L..Y., et al.: “NumericalSimulation of Ekofisk ReservoirCompaction and Subsidence: Treating the MechsnicaI Behaviorof the Overburden and Reservoir,” Proc., Eurock ’94, tikMechanics in Petroleum Engineering, Delft, Netherlands (1994)787.

3BZ

.


6.

7.

8.

9.

10.

II.

12.

13.

14,

PIiscbke, B.: ~mite Hemcnt Analysis of Compaction andSubsidence - Experience Gained from Several Chalk Fields,”Prac., Eurock ’94, Rock Mwhanics in Petroleum Engineering,Delft, Netherlands (1994) 795.W, M. J.: Analysis of the Valhall Oi@eld Subsidence,Sediment Deformation Research Center, London (Nov. 1987).Hibbitt. Kadsson, and Soznsen, Inc., ABAQUS User’s Manual,Version 5.6 (1996).Fjaer, E. et al.: Petroleum Related Rock Mechanics, Elsevier,New York (1992).Andersen, M. A.: Petroleum Research in North Sea Chalk,JointChaIk Research Phase IV (1995).Kristiansen, T. G.: -omechanical Characterization of theOverburden Above the Compacting ChaIk Reservoir at Valhall,”Proc., Eumk ’98, Rwk Mechanics in Petroleum Engineering,Trondheim, Norway (1998).Berry, D. S.: “An Elastic Treatment of Ground Movement dueto Mining -1. Isotropic Ground,” J. A4ech. Phys. Solids ( 1960) 8

~, D. S. and Sales, T. W.: “An Elastic Treatment of GroundMovement due to Mining -II. Transversely Isotropic Ground,”J. Mech. Phys. Solid, (1961) 952.Berry, D, S. and Sales, T. W.: “An Elastic Treatment of GroundMovement due to Mining -~. Three Dimensional Problem,Transversely Isotropic Ground,” J. Mech. Phys. Solids (1962) 1073.

Appendix - The ABAQUS Cap Plasticity ModelMaterial behavior for aIl chaIk and overburden elements wasdetermined by in house correlations, where elastic behaviorwas assumed to be isotropic (chalk) or anisotropic(overburden), and inelastic behavior was described using theABAQUS cap plasticity modeI.The A8AQUS cap plasticity models combines a Drucker-Prager failure surface with an elliptical cap ~lg. A-I) in p-fspace,

~fm = (p- p.) + (Ktt) - R(cf+ p. tan~), .......(A-l)

where

t=;[l+;-[l-;](:y] ....................................(A-2)

q is the Mi.ses equivalent stress, ~ is the third stress invariant,

()~ = tan-~ MDP , /? controls the shape of the ellipse and

Kt =R

l+a’-a’sec~. .........- ..................................(A-3)

A surface characterized by the parameter a’ effects asmooth transition between the Drucker-Prager surface and theelliptical cap,

Yielding on the elliptical cap results in work hardening,

P, =Fcp%) , ...........................................................(A-4)

where

Pc - Rd

“=l+Rtanfl. ...................................................... (A-5)

No functional form is stipulated for this relation. Rather, “pc is entered as a piecewise linear function of volumetricplastic strain.

Plastic flow is defined using a separate flow potentiaIfunction that is associated in the deviatoric (p) plane,associated along the elliptical cap in the meridional (p-~plane, and nonassociated along the Drucker-Prager ftihrresurface and @ansition circle in the meridioti plane. menonassociated flow potential surface is elliptical and isconstrained such that it forms a continuous and smoothpotential surface with the elliptical cap.

S1Metric Conversion Factorsfi X 3.048* E-01 - m

lbf X 4.448222 E+OO= Nlbtigal x 1.198264 E+02 - kg/m3

psi x 6.894757 E-03 - MPa“Conversion factor is exact.

~Formation De~th to TOD (ml

Pliocene/Miocene 112.7Lower Miocene 1437.11Middle Eocene 2190.29

Table 2- Elastic Constants for Overburden used inCurrent Study

Formation ConstantsPliocene/Miocene E“ = 775 MPa, Eh = 870 MPa,

~h”= ~h = 0.46, Ghv= 320 MPaLower Miocene E“. 590 MPa, Eh. 650 MPa,

vh”= vh = 0.45, Ghv= 240 MPaMiddle Eocene E.= 720 MPa, Eh = 810 MPa,

vh”= ~h = 0.46, Ghv= 300 MPaPaleocene E.= 730 MPa, Eh. 820 MPa,

vh”= vh = 0.44, Gh. = 310 MPaUnderburden E“. 1980 MPa, Eh = 2390 MPa

vh”= vh = 0.30, Ghv= 840 MPa

Table 3- Drucker-Prager Constants for Overburdenand Underburden used in Current Study

Formation Drucker-Praae r ParametersPliocene/Miocene ~= 37.9, d = 7.79 MPa, k = 0.794

Lower Miocene P= 34.4, d= 6.85MPa, k= 0.814Middle Eocene ~= 37.0, d= 7.53 MPa, k = 0.799

Paleocene ~= 37.3, d = 7.63 MPa, k = 0.797Underburden B = 47.8, d= 11.03 MPa, k= 0.778

383

8 P. D. PATTILLO, T. G. KRISTIANSEN, G. V. SUND, R. M. KJELSTADLI SPE 47274

Table 4- Penetration of the Overburden andUnderburden Yield Surfaces by a Uniaxial Load Path

Siope pat Cap q at Capof Path intersection intersection

~~~ ~Pliocene/Miocene

Lower Miocene 1i.7 2:60 0:539Middle Eocene 9.3 4.70 0.773

Paleocene 14.0 4.54 1.14Underburden 42.7 8.77 8.10

Fig. I.-overall view of ffnlte element mesh. The SIX reservoirIsysm and Palecoarre overburden are very thin and are notdiscernible on this figure, bur appear as a single thick ls~r. Theoutsmoat rtng of eIementa are esmi-inflnlts boundsry elementsand do not overfav tfro reaarvolr simulstor mesh. me nletforrn

~.~!.?x.!?.p.qg~~ in ~.? ~!e~.of ths mesh.

VK&>~]

: j+~...; . ......... ..... .:. ..... ....!..... ...........

....................... .... ..... .!....!.

Fig. 2.-Plan view of reservoir Mrtlon of flnlte element mesh, A

30

25

20

& 15

10

5

0

,’; — Plio/Miccene

; ---- Lower Miocene

~’ .--. --- Middle Eocene

t

/’ -----Underburden

f“8:,.

. .--.—-

0 0.1 0.2 0.3 0.4 0.5

-I?Pvol

Fig. 3.-Oap hardening bshsvior for overburden and,,.*rhur*n.

! 0.35 A

0.30

o~~ ~

o 10 20 30 40

OB Streae (MPa)

Fig. 4.-Reaulte of fitting ABAQUS cap model to type curves ofAnderasn’O. Solid lines are predicted response. Symbols areoriginal type curves.

cc-~ur of TVD to top of the Tor formation Is overlald to Indlcetsthe pcsltion of the mesh mlatfve to the extant of the reservoir.

384

.


7T

6- -.

5- - .E

I ~

4

a32- -

1- -

0 I

1980 1990 2000 2010 2020 2030 2040

Year

Fig. 5.4omparfaon of subsidence pradictlons at the Valhallplatform comp[ax with field maaauramenta. The Plaatlc OB(Overburden) curve la the currant etudy.

0.4

I+

0.35 t+A,

E ‘“3$0.25

80.2

1

0.15

0.1@

0.05 I

o~19s0 1985 1990 1995 2000

Year

Fig. 6.-Oomparfaon of eubsidenca rata pradictiona at the Valhallplatform complex wIM field maaauramenta. The Plaatic OB(Overburden) curve Is the currant study.

8

7

6

3

2

1

0

-.\s,+

.T\\., \\

‘\ \\{.‘. \\

\ \\\,,\

\

\\\,‘\\

\,i,:

0 1 2 3 4p, MPa

Fig. 7.--Typlca1 atraaa patha In the Valhall overburden.

o

-1

E -2

1

-3

-4

=-5$g%

g -7

84

-9

5

I

-1040 6671

Diatanm Along Peth, m

Fig. 8.-Oompariaon of compaction and eubeldenca predictionsfor two models with different elestlc properties in the overburden.

Fig. 9.—Vertical dlsplacament at top of raaervolr on December,2031. Only the region overlaying the reaarvoir almulator mesh lashown. Contour Ievela are meters.

385

10 P. D. PAITILLO, T. G. KRISTIANSEN, G. V. SUND, R. M. KJELSTADLI—. SPE 47274

Fig 10.-VerWcal dlaplacement at mudline on Dacambar, 2031.Only tie region ovarIaying the raaarvoir simulator maah isshown. Contour Ievela are matara. Dleplacement at platform la&3 matera.

. . ..~ iiiiii~it+{ 44t.,..

t

k’MDP ~ Transition Surface

s-a~

..VI,,,ipsfiu

-..

~R(d+p~tan~)~Fig. A-i.-ABAQUS ylald aurfaca.

366

Documents

Reservoir Compaction and Seafloor Subsidence at Valhall