17
ABSTRACT Presently available techniques for predicting quantitative reservoir quality typically are limited in applicability to specific geographic areas or litho- stratigraphic units, or require input data that are poorly constrained or difficult to obtain. We have developed a forward numerical model (Exemplar) of compaction and quartz cementation to provide a general method suited for porosity prediction of quartzose and ductile grain-rich sandstones in mature and frontier basins. The model provides accurate predictions for many quartz-rich sand- stones using generally available geologic data as input. Model predictions can be directly compared to routinely available data, and can be used in risk analysis through incorporating parameter optimiza- tion and Monte Carlo techniques. The diagenetic history is modeled from the time of deposition to present. Compaction is modeled by an exponential decrease in intergranular vol- ume as a function of effective stress. The model is consistent with compaction arising from grain rearrangement, ductile grain deformation, and brit- tle failure of grains, and accounts for the effects of fluid overpressures and stable grain packing con- figurations. Quartz cementation is modeled as a precipitation-rate–controlled process according to the method of Walderhaug (1994, 1996) and Walderhaug et al. (in press). Input data required for a simulation include effective stress and temperature histories, together with the composition and texture of the modeled sandstone upon deposition. Burial history data can 433 AAPG Bulletin, V. 83, No. 3 (March 1999), P. 433–449. ©Copyright 1999. The American Association of Petroleum Geologists. All rights reserved. 1 Manuscript received August 14, 1997; revised manuscript received June 19, 1998; final acceptance October 5, 1998. 2 Geologica, P.O. Box 8034, N-4003 Stavanger, Norway; e-mail: [email protected] 3 Statoil, 4035 Stavanger, Norway; e-mail: [email protected] We thank Esso Norge a.s and Exxon Production Research Company for providing the funding to implement and test Exemplar, and Stan Paxton, David Awwiller, et al., from Exxon Production Research for their valuable guidance in this work. Finally, we thank Linda Bonnell, Sal Bloch, Alton Brown, and Dick Larese for their thorough reviews of earlier versions of this manuscript. Predicting Porosity through Simulating Sandstone Compaction and Quartz Cementation 1 R. H. Lander 2 and O. Walderhaug 3 be obtained from basin models, whereas sandstone composition and texture are derived from point- count analysis of analog thin sections. Exemplar predictions are consistent with measured porosity, intergranular volume, and quartz cement fractions for modeled examples from the Quaternary and Tertiary of the Gulf of Mexico Basin, the Jurassic of the Norwegian shelf, the Ordovician of the Illinois basin, and the Cambrian of the Baltic region. INTRODUCTION Reservoir quality is one of the important uncer- tainties in wildcat drilling (Bloch, 1994a; Wilson, 1994). Present approaches to reservoir quality prediction, however, commonly are limited in applicability, are difficult to apply, or are of unproven accuracy. The need for improved methods has motivated us to develop a model, known as Exemp- lar, that is designed to •Consider the most significant porosity control- ling processes in sandstone lithologies that are common hydrocarbon reservoirs •Make predictions that approach the measure- ment accuracy of available data •Use input data that are commonly available, eas- ily obtained, or readily estimated in both mature and frontier basin settings •Produce predictions that can be compared directly to petrographic thin sections •Include a rigorous approach to uncertainty assessment so that reservoir quality predictions can be stated in probabilistic terms •Be fast and easy to use on personal computers available to explorationists In our initial efforts we have targeted quartz-rich sandstones because they are the most common sandstone reservoir type and because their porosity commonly is controlled by just two classes of dia- genetic processes: compaction and quartz cemen- tation. In this paper, we review the model’s design and algorithms, give conceptual justifications for the approaches we have used, and present some

Predicting Porosity through Simulating Sandstone ...carbcap.geos.ed.ac.uk/rsh-journal-downloads/Predicting-Porosity... · Predicting Porosity through Simulating Sandstone Compaction

Embed Size (px)

Citation preview

ABSTRACT

Presently available techniques for predictingquantitative reservoir quality typically are limited inapplicability to specific geographic areas or litho-stratigraphic units, or require input data that arepoorly constrained or difficult to obtain. We havedeveloped a forward numerical model (Exemplar)of compaction and quartz cementation to provide ageneral method suited for porosity prediction ofquartzose and ductile grain-rich sandstones inmature and frontier basins. The model providesaccurate predictions for many quartz-rich sand-stones using generally available geologic data asinput. Model predictions can be directly comparedto routinely available data, and can be used in riskanalysis through incorporating parameter optimiza-tion and Monte Carlo techniques.

The diagenetic history is modeled from the timeof deposition to present. Compaction is modeledby an exponential decrease in intergranular vol-ume as a function of effective stress. The model isconsistent with compaction arising from grainrearrangement, ductile grain deformation, and brit-tle failure of grains, and accounts for the effects offluid overpressures and stable grain packing con-figurations. Quartz cementation is modeled as aprecipitation-rate–controlled process according tothe method of Walderhaug (1994, 1996) andWalderhaug et al. (in press).

Input data required for a simulation includeeffective stress and temperature histories, togetherwith the composition and texture of the modeledsandstone upon deposition. Burial history data can

433AAPG Bulletin, V. 83, No. 3 (March 1999), P. 433–449.

©Copyright 1999. The American Association of Petroleum Geologists. Allrights reserved.

1Manuscript received August 14, 1997; revised manuscript received June19, 1998; final acceptance October 5, 1998.

2Geologica, P.O. Box 8034, N-4003 Stavanger, Norway; e-mail:[email protected]

3Statoil, 4035 Stavanger, Norway; e-mail: [email protected] thank Esso Norge a.s and Exxon Production Research Company for

providing the funding to implement and test Exemplar, and Stan Paxton,David Awwiller, et al., from Exxon Production Research for their valuableguidance in this work. Finally, we thank Linda Bonnell, Sal Bloch, AltonBrown, and Dick Larese for their thorough reviews of earlier versions of thismanuscript.

Predicting Porosity through Simulating SandstoneCompaction and Quartz Cementation1

R. H. Lander2 and O. Walderhaug3

be obtained from basin models, whereas sandstonecomposition and texture are derived from point-count analysis of analog thin sections. Exemplarpredictions are consistent with measured porosity,intergranular volume, and quartz cement fractionsfor modeled examples from the Quaternary andTertiary of the Gulf of Mexico Basin, the Jurassic ofthe Norwegian shelf, the Ordovician of the Illinoisbasin, and the Cambrian of the Baltic region.

INTRODUCTION

Reservoir quality is one of the important uncer-tainties in wildcat drilling (Bloch, 1994a; Wilson,1994). Present approaches to reservoir qualityprediction, however, commonly are limited inapplicability, are difficult to apply, or are of unprovenaccuracy. The need for improved methods hasmotivated us to develop a model, known as Exemp-lar, that is designed to

•Consider the most significant porosity control-ling processes in sandstone lithologies that arecommon hydrocarbon reservoirs

•Make predictions that approach the measure-ment accuracy of available data

•Use input data that are commonly available, eas-ily obtained, or readily estimated in both matureand frontier basin settings

•Produce predictions that can be compareddirectly to petrographic thin sections

•Include a rigorous approach to uncertaintyassessment so that reservoir quality predictions canbe stated in probabilistic terms

•Be fast and easy to use on personal computersavailable to explorationists

In our initial efforts we have targeted quartz-richsandstones because they are the most commonsandstone reservoir type and because their porositycommonly is controlled by just two classes of dia-genetic processes: compaction and quartz cemen-tation. In this paper, we review the model’s designand algorithms, give conceptual justifications forthe approaches we have used, and present some

example simulations. The model provides reason-ably accurate reservoir quality predictions forquartzarenites, sublitharenites, and subarkoses, andhas proven to be a useful predictive tool for bothmature and frontier regions. The model alsoappears to be well suited to porosity prediction ofductile grain-rich rocks, but more geologic data setsare needed for calibration before it can be confi-dently applied to reservoir quality prediction insuch lithologies.

Exemplar simulates the evolution of sandstoneporosity and composition throughout the geologichistory of the modeled unit. In addition, it simulatesthe rates of porosity reduction due to compactionand cementation through time (output data typesare given in Table 1). Input data needed to conducta simulation include a description of the texture andcomposition of the sandstone upon deposition,burial history information, and parameter values forthe compaction and quartz cementation algorithms.Depositional sandstone texture and compositionaldata are derived from standard petrographic analy-ses of reservoir samples from nearby wells or fromlithologic analogs in frontier areas. Basin modelingresults for prospect reservoir intervals provide thenecessary temperature and effective stress historyinput (burial depth can be substituted for effectivestress for areas that have not experienced signifi-cant fluid overpressures). Finally, the appropriatecompaction and quartz cementation parameterscan be obtained from calibration studies. A listing of

input data types that can be used by the model isgiven in Table 2.

In addition to its utility for porosity predictionprior to drilling, Exemplar has proven to be a use-ful paleothermal indicator (Awwiller and Summa,1997; Lander et al., 1997a, b). Because quartzcementation is strongly controlled by burial history,the model can be used to constrain burial histo-ries by comparing model predictions with mea-sured values. Exemplar also may provide a moreaccurate depiction of reservoir quality variationsand heterogeneties within fields than do pure geo-statistical methods when it is used in concert withupscaling techniques; furthermore, the modelshould provide a more accurate basis for assessingthe evolution in properties of sandstones that act ashydrocarbon carrier systems than do current basinmodeling systems.

EXISTING APPROACHES TO RESERVOIRQUALITY PREDICTION AND COMPARISONWITH EXEMPLAR CONCEPTUAL FRAMEWORK

Existing reservoir quality models tend to fall intotwo categories (Wood and Byrnes, 1994): effect-oriented models, such as statistical correlations ofporosity with other variables, and process-orientedmodels, such as geochemical reaction-path modelsthat are based on the thermodynamics and kinetics ofminerals, aqueous species, and gases. The statistical

434 Predicting Porosity

Table 1. Exemplar Output Results for Each Model Time Step

Rock Fractions PorosityQuartz cementNonquartz cementIntergranular volumeQuartz grainsNonquartz framework grainsMatrix

Rock Volumes (cm3) Bulk rock volumeQuartz cement (for current time step)Quartz cement (cumulative)Nonquartz cement (cumulative)

Rates (cm3/m.y.) Quartz cementationNonquartz cementation Compaction

Porosity Controls Copl—absolute porosity loss due to compaction (Ehrenberg, 1989)Cepl—absolute porosity loss due to cementation (Ehrenberg, 1989)ICOMPACT: copl / (copl + cepl) (Lundegard, 1991)

Other Quartz surface area (cm2)Average overgrowth thickness (mm)%Ro (Sweeney and Burnham, 1990)

approach can be accurately applied to sandstoneswith less than 10% cement, but this approachbreaks down for more highly cemented sandstones(Bloch, 1991). An additional shortcoming to thestatistical approach is that accurate model predic-tions are constrained to sandstone compositions,textures, and geologic settings represented by thesamples included in calibration data sets (Bloch andHelmold, 1994). Thus, it is difficult to apply theempirical approaches to frontier areas with little orno data, to mature areas where statistical studieshave yet to be undertaken, or to depth ranges out-side that of an existing calibration data set.

Geochemical models, such as those reviewed byMeshri (1990) and Wood (1994), are appealingbecause by using a first-principle approach to sim-ulating diagenetic reactions they should ideally bemore broadly applicable than empirical methods.Although reaction-path models provide importantinsights into diagenetic processes, in practice thepresent generation of models is difficult to applyto quantitative porosity prediction. Many suchmodels ignore compaction, frequently the singlegreatest cause of porosity reduction (Lundegard,1991), as well as the effect of diagenesis on thesurface area of reactive minerals. These models suf-fer from substantial uncertainties in kinetic andthermodynamic constants because many phaseshave not yet been characterized, and the extent towhich the existing data can be extrapolated fromlaboratory conditions to geologic time scales andenvironments is not always clear. Finally, mostreaction-path models do not explicitly distinguishbetween the detrital or authigenic occurrence ofminerals, making it difficult to compare predic-tions with petrographic data.

The approach that we have taken to reservoirquality prediction attempts to synthesize effect-oriented and process-oriented methods. Althoughthe model predicts the result of diagenetic process-es, it does not employ a first-principle approach tosimulating these processes. Instead, our “hybridsimulator” [terminology of Wood and Byrnes(1994)] employs process algorithms with inputparameters that are empirically calibrated to geologi-cal data sets. The advantage of using this approach istwo-fold. First, by calibrating individual processes,the model should have wider applicability than thestandard statistical approach and can be based ondata reflecting geologic time durations and environ-ments rather than laboratory conditions. Second, thecomputational efficiency of a hybrid model can beimproved by orders of magnitude compared to afirst-principles approach. Greater computational effi-ciency is important because it permits predictiveuncertainties to be evaluated much more rigorously.A thorough investigation of uncertainties is a prereq-uisite to quantitative, probabilistic approaches toreservoir quality prediction.

A hybrid approach to porosity modeling wasalso taken by Waples and Kamata (1993) for a vari-ety of lithologic types, including sandstones.Similar to Exemplar, the model used by Waples andKamata is calibrated against geological data setsand is more widely applicable than the empiricalmodels it was designed to replace. Although thisapproach succeeds in its goal to improve on themethods used by basin models, it was not designedfor reservoir quality prediction (Waples andKamata, 1993). Consequently, it does not take intoconsideration the important effects that variationsin texture and composition have on porosity. By

Lander and Walderhaug 435

Table 2. Input Parameters for Exemplar Simulations

Initial Porosity (%)Sediment Quartz framework grain (%)Composition Nonquartz framework grain (%)

Matrix (%)Average grain diameter (mm)

Compaction IGVf —stable packing intergranular volume (%)Parameters β—exponential rate of intergranular volume loss with effective stress (1/MPa)

Quartz cement abundance sufficient to arrest compaction (%)

Quartz Quartz precipitation rate preexponential constant (mol/cm2s)Cementation Quartz precipitation rate exponential constant (1/°C)Parameters Minimum temperature necessary for the onset of quartz precipitation (°C)

Burial History Temperature (°C), time (m.y.)Burial depth (m), time (m.y.)Effective stress (MPa), time (m.y.) (or use burial depth and assume hydrostatic pressure)Grain coat completeness (%), time (m.y.)Nonquartz cements (cm3), time (m.y.)

contrast, Exemplar uses sandstone textural andcompositional data as input. In the present version,the model targets quartz-rich and ductile grain-richsandstones because the majority of producing sand-stone reservoirs are represented by these litholo-gies, and because the primary controls on porositycan be linked to compaction and quartz cementa-tion. In addition, the model explicitly predictsindices of compaction and cementation that can bedirectly compared with petrographic analyses ofthin sections, making it straightforward to assessthe predictive accuracy of the results, as well as tocalibrate model parameters.

MODEL DESIGN

Exemplar is designed to predict porosity in sand-stones where porosity is dominantly controlled bycompaction and quartz cementation. This porosityprediction is directly comparable to petrographical-ly determined intergranular porosity, but not tocore analysis porosity, which also may contain sec-ondary porosity, fracture porosity, and microporosi-ty (Pittman, 1979).

Controls on Compaction

Compaction in sandstone is the reduction inbulk rock volume that occurs in response to fourclasses of processes (in the generally acceptedorder of importance): grain rearrangement, plasticdeformation, dissolution, and brittle deformation(Wilson and Stanton, 1994). In our assessment, amodel relating the effects of grain rearrangement,plastic deformation, and brittle deformation toeffective stress should provide an adequate basisfor simulating compaction in most quartzose andductile grain-rich sandstones.

We have disregarded grain-to-grain “pressure”dissolution in the present version of the modelbecause we see little empirical evidence of it beinga primary control on reservoir quality for the target-ed group of sandstone lithologies. Quartzose hydro-carbon reservoirs typically show little evidence forextensive dissolution of quartz except where clayrims occur (Heald, 1956; Thomson, 1959; Wilson,1984; Houseknecht, 1988) or at stylolites (Sippel,1968; Land and Dutton, 1978; Suchecki and Bloch,1988; Ehrenberg, 1990; Paxton et al., 1990;Ajdukiewicz et al., 1991; Bloch, 1991; Szabo andPaxton, 1991; Bjørkum, 1994). Although grain dis-solution at stylolites is common, such dissolutionhas little effect on rock properties of reservoirintervals at the thin-section or core-plug level.

The significance of secondary porosity causedby feldspar and rock fragment dissolution for

reservoir quality remains controversial. Giles andde Boer (1990) and Bloch (1991, 1994b, c), in theirreviews of the topic, concluded that the signifi-cance of secondary porosity commonly is overstat-ed; furthermore, they point out that where sec-ondary porosity does occur it has little effect onthe accuracy of porosity predictions because it iscounterbalanced by the local reprecipitation of thedissolved material.

Grain rearrangement leads to compaction whenframework grains move into tighter packing config-urations, and correlates with the stress transmittedthrough framework grains. A number of petrogra-phers regard grain rearrangement as the primarycontrol on compaction of well-sorted sandstonesthat are poor in matrix and ductile grains. Exxonworkers reported that a stable packing configura-tion represented by an intergranular volume (IGV)of 26% is reached through grain rearrangement byapproximately 2 km burial depth for well-sortedsamples (Paxton et al., 1990; Ajdukiewicz et al.,1991; Szabo and Paxton, 1991). Other workershave suggested that such a stable packing configu-ration cannot be achieved without some minor dis-solution at grain contacts sufficient for grains toslip past each other (e.g., Füchtbauer, 1967).Palmer and Barton (1987) contended that rear-rangement reduces IGV to a minimum of 34%, andthat further compaction requires some grain disso-lution. Siever and Stone (1994) asserted that grainrearrangement accounts for an IGV decline toapproximately 31%, and that a stable packing con-figuration of 24% IGV requires minor pressure solu-tion. Although there is some disagreement as to theminimum IGV attained through grain rearrange-ment alone, most workers agree that the process isa major factor in controlling compaction, and thatit can be related to effective stress.

Plastic deformation of grains takes place whengrains deform under stress and is the primary con-trol on compaction in ductile grain-rich rocks(Pittman and Larese, 1991). Bulk compactionoccurs in response to plastic deformation becauseit leads to consolidation as pores collapse or areinvaded by deforming grains, and because it pro-motes additional grain rearrangement. The extentof compaction due to ductile grain deformation is afunction of the abundance and ductility of ductilegrains and the effective stress (Pittman and Larese,1991). A compaction model based on correlationswith effective stress should provide a good basis forsimulating the effects of plastic deformation solong as the model can take into account the effectof differing amounts and types of ductile grains.

Brittle grain deformation commonly is thought toplay only a minor role in compaction (Wilson andStanton, 1994). Recent studies employing cathodo-luminescence techniques suggest the significance of

436 Predicting Porosity

brittle grain failure, however, may be underestimat-ed in some settings. Using cathodoluminescence,we have observed numerous healed fractures inquartz grains from Miocene sandstones from theGulf of Mexico. Similar features in quartzose sand-stones have been reported from other regions(Dunn, 1994; S. Bloch, 1994, personal communica-tion; Laubach, 1997). Because brittle grain failureoccurs in response to grain-to-grain stress, effectivestress should provide a suitable basis for a com-paction model incorporating this process.

Simulation of Compaction

A hybrid compaction model should be able toaccount for the salient points discussed if com-paction is modeled as a function of effective stress,and if the effects of grain ductility, sorting, and sizeare taken into account. A requirement for such amodel is that it should provide predictions that canbe directly compared to an index of sandstonecompaction so that the model can be calibratedand so that uncertainty in model predictions can bequantitatively assessed.

A difficulty in constructing and testing com-paction models is that there are no known directmeasures of the extent of compaction in sand-stones (Wilson and Stanton, 1994). As a result,compaction can be evaluated in sandstones only byusing indirect measures and by assuming a deposi-tional porosity based on analogs. Examples of indi-rect measures of compaction that have been devel-oped include a contact index (Taylor, 1950), a tightpacking index (Wilson and McBride, 1988), and anintergranular volume (IGV) (Paxton et al., 1990;Ajdukiewicz et al., 1991; Szabo and Paxton, 1991),which are defined, respectively, as the averagenumber of intergranular contacts per grain; theaverage number of long, concavo-convex, andsutured contacts per grain; and the sum of inter-granular porosity and cements. Although the con-tact and tight packing indices are useful measuresof compaction, McBride et al. (1990) reported thatthe measures are highly subjective and that theycannot be derived from standard point-count analy-ses of thin sections. We use IGV as an indirect mea-sure of compaction for the model because it ideallyshould be less prone to measurement subjectivity,can be determined from standard point-count data,and can be used directly in the simulation of inter-granular porosity when cement volumes are alsoknown.

Existing porosity vs. effective stress (or depth)functions provide a good basis for building a com-paction model. Empirical functions incorporatingburial depth have been used for accurate porositypredictions for sandstones with less than 10%

cements so long as they also incorporate sandstonetextural and compositional parameters (Bloch,1991). Porosity decline functions (e.g., Athy, 1930)have been modified for use in basin modeling sys-tems to account for the effects of fluid overpres-sures by substituting effective stress for burialdepth (Ungerer et al., 1990; Schneider et al., 1994).We make two additional modifications to adapt theapproach to simulation of the effects of sandstonecompaction. IGV, an indirect index of compaction,is substituted for porosity, and we introduce a termrepresenting the stable packing configuration ofsandstones. The resulting compaction function is

(1)

where IGV is the sum of pore space, cements, andmatrix material (volume fraction); IGVf is the sta-ble packing configuration (volume fraction); φ0 isthe depositional porosity (volume fraction); m0 isthe initial proportion of matrix material (volumefraction); β is the exponential rate of IGV declinewith effective stress (MPa–1); and σes is the maxi-mum effective stress (MPa).

Because matrix material generally has little or nocompressive strength, IGV here is defined asincluding matrix material and intergranular porespace and cement. We also assume that matrixmaterial in grain-supported sandstones does notcompact significantly, and that no cements occur atthe time of deposition of the sand when the effec-tive stress is zero.

The frame of reference for the model is a 1-cm3

volume defined by uncompacted sediment at thedepositional surface. This reference frame is compa-rable in scale to thin sections and core plugs, thebasic data sources for evaluating and calibrating sim-ulation results. The bulk compaction of the rockcan be obtained from the following expression:

(2)

where v is the bulk volume of the compacted rock(cm3), v0 is the bulk rock volume upon deposition(cm3), IGV is the sum of intergranular pore space,cements, and matrix material in the current timestep (volume fraction), φ′0 is the porosity upondeposition (volume fraction), and m′0 is the matrixmaterial upon deposition (volume fraction).

Although the model conserves the volumes anddensities of detrital materials, it does not attempt tomaintain mass balance of pore fluids or cements.Through the course of a simulation, the net volumeof pore fluid typically is reduced, whereas the netvolume of solids increases. The modeled compaction

v vm

IGV= − ′ − ′

− ′

0

0 01

1

φ

IGV IGV m IGV ef fes= + + −( ) −φ βσ

0 0

Lander and Walderhaug 437

applies only to the frame of reference and not nec-essarily to the depositional unit as a whole. Whencementation occurs, the overall unit compactionexceeds that of the sample due to dissolution in thesource regions for quartz cement (stylolites).

Because sandstones exhibit negligible decom-paction with unloading at the model scale of obser-vation, compaction is not reversible in the modeleven in cases where effective stress decreases, suchas after uplift and erosion. Compaction will also ter-minate when a specified amount of quartz cementhas precipitated, strengthening the sandstoneframework. The strengthening effect of quartzcementation may be partly responsible for theoccurrence of abundant quartz cement in somesandstones with anomalously high IGV values (e.g.,>30%). Minor amounts of quartz cement that pre-cipitate prior to achievement of a stable packingconfiguration could retard compaction until deeperburial provides sufficiently high temperatures forquartz cements to fill remaining pore space. Such ascenario is consistent with compaction experi-ments of Pittman and Larese (1991), where theyevaluated the effect of small amounts of quartzcement on the mechanical strength of ductile grain-rich sandstones. In their experiments, they used astarting material comprised of 22% green shalerock fragments, 40% quartz grains, and 38% porosi-ty. In the absence of quartz cement, the samplecompacted to an IGV of 8.8% at an effective stressof 51.7 MPa. An otherwise identical sample with5.5% quartz cement artificially grown prior to com-paction was more resistant to compaction, main-taining an IGV of 19.3% at 51.7 MPa.

Cementation

Quartz CementationA number of quartz cementation mechanisms

have been proposed for quartzose sandstones,including coupled cementation and compactionarising from pervasive intergranular pressuresolution, precipitation of quartz cements fromcooling fluids, and diffusion-controlled derivationof quartz cement from nearby shales. Previousmodels of quartz cementation based on thesemechanisms cannot account for the lack of signif-icant intergranular dissolution that occurs in mostquartz-cemented sandstones that are free of clay-coated grains, call upon unrealistically extensivefluid f luxes, or predict more pronounced gradi-ents in quartz cement abundance away fromcement sources than has been observed in typicalreservoir sandstones (Walderhaug, 1996; Walder-haug et al., in press).

By contrast, Exemplar incorporates a model ofquartz cementation that assumes that (1) the

quartz cement is derived from nearby stylolites orprotostylolites (i.e., regions where quartz grainscome in contact with clay-rich zones such as claylaminae or silty interbeds), (2) of the three stepsthat control the quartz cementation process, sili-ca dissolution, diffusion/advection, and precipita-tion, silica precipitation is the rate-limiting orcontrolling step, (3) the surface area available forquartz cement precipitation is a function of thesize, fraction, and extent of coating of quartzgrains, as well as of the overall rock porosity, and(4) no quartz dissolution occurs within the frameof reference for the simulation. The model hon-ors petrographic (cathodoluminescence) evi-dence showing little evidence of extensive inter-granular dissolution (Suchecki and Bloch, 1988).In the following section, we brief ly review theimportant elements of the quartz cementationmodel and its implementation in Exemplar. Formore detail on how the model was derived andhow it compares with other models of quartzcementation, see Walderhaug (1994, 1996) andWalderhaug et al. (in press).

The essential elements of the quartz cementa-tion model are the kinetics of quartz precipitationand the surface area available for quartz cementgrowth. The rate of quartz cementation per unit ofsurface area has been shown empirically to be afunction of temperature (Walderhaug, 1994):

(3)

where a is the quartz precipitation rate preexpo-nential constant (mol/cm2 s), b is the quartz precip-itation rate exponential constant (°C–1), and T istemperature (°C).

This function can be extended to calculate thetotal amount of quartz cement precipitated duringan increment in time by taking into account thesurface area available in the sandstone for precipita-tion of quartz cement, and by considering the tem-perature range experienced by the sample

(4)

where qcv is the volume of quartz that precipitates(cm3), m is the molar weight of quartz (60.08g/mol), ρ is the density of quartz (2.65 g/cm3), A isthe quartz surface area (cm2), t is the duration ofthe time step (m.y.) converted to seconds, a is thequartz precipitation rate preexponential constant(mol/cm2 s), b is the quartz precipitation rate expo-nential constant (1/°C), and cn and dn are constantsfor each time step n (derived from the sample’stemperature history).

qcvm

Aa dtb cnt dn

t

= +( )∫ρ10

0

r abT= ( )10

438 Predicting Porosity

Quartz surface area plays an important role incontrolling the net rate of quartz cementation, asshown in equation 4. Quartz surface area in themodel is a function of the abundance of detritalquartz grains in the initial sediment, the averagequartz grain size, and the porosity through time:

(5)

where A is the quartz surface area for the presenttime step (cm2), qgf0 is the abundance of quartzgrains in initial sediment (fraction), v0 is the initialrock volume (cm3), D is the average diameter of ini-tial quartz grains (cm), φ is the porosity for thepresent time step (fraction), φ0 is the initial porosi-ty (fraction), and coat is the quartz surface area thatis coated and thus unable to act as a substrate forfurther quartz precipitation (fraction).

The surface area definition assumes that thequartz grains are spherical and are of uniform size.Surface area is modeled to decrease as a function ofporosity to account for the effects of compactionand cementation (Figure 1). Compaction reducesquartz surface area by increasing grain contactarea, as well as by the injection of matrix materialinto pore spaces. Cementation can cause surfacearea reduction when quartz grains are encased bypore-filling cements. The model also assumes thatcoatings on quartz framework grains reduce sur-face area proportionally to the coated area. Thecoat variable in equation 5 refers to the proportionof quartz surface area that no longer permits nucle-ation of overgrowths because of the occurrence ofgrain-coating materials. Grain coatings are associat-ed with detrital materials, such as clays and Fe-oxy-hydroxides, as well as with authigenic phases, suchas clay minerals, carbonate (siderite), or microcrys-talline quartz.

Nonquartz CementationThe model permits the occurrence of cements

other than quartz to be defined through time.Although the model does not yet simulate the pre-cipitation of such cements, it does consider theeffect that they can have on compaction, quartzprecipitation, and porosity. Occurrence of non-quartz cements can reduce the extent of com-paction by filling pore space when modeled IGVvalues exceed the stable packing configuration. Inaddition, nonquartz cement can reduce the abun-dance of quartz cement by reducing available porespace and by retarding the net rate of quartzcementation by reducing the available surface area.The model permits the amount of nonquartzcements to vary with time.

A coatqgf v

D= −( )

1

6 0 0

0

φφ

Overview of the Simulation Sequence

After the parameter values and model optionsare defined and the simulation is initiated, the pro-gram begins by reading input parameters and bymaking various initial calculations, such as unitconversions and rock component volume determi-nations. The model then begins to step forwardthrough time using time-step durations that areselected dynamically based on the rate of changein modeled porosity. Time steps are typically noless than 105 yr. The calculations for a given timestep are done in the following sequence: com-paction, precipitation (or dissolution) of nonquartzcements, quartz surface area, and quartz cementvolume. The compaction and cementation calcula-tions are not directly coupled because compactionmost likely adjusts more rapidly to changes in buri-al conditions than cementation, and because com-paction commonly dominates the early diageneticevolution of sandstones. Typical execution timesare on the order of seconds for desktop computersfor a simulation ranging from the time of burial tothe present day.

MODEL CALIBRATION

Compaction

Our compaction model contains three terms thatreflect the texture and composition of sandstones:φ0 (depositional porosity), IGVf (stable packing con-figuration), and β (exponential rate of compactionwith effective stress). Values for the φ0 term can betaken from measurements of near-surface sandstoneporosities or artificially mixed and packed sand-stones. Although the depositional porosity is largelyindependent of sandstone composition, it is strong-ly controlled by texture. Well to moderately sortedf luvial and beach sandstones have depositionalporosities of 47.5 ±3.2% (Atkins and McBride,1992), but more poorly sorted samples have valuesthat are as much as 10–20% lower due to greaterpacking efficiency of grains (Beard and Weyl, 1973;Clarke, 1979; Dickinson and Ward, 1994).

The IGVf and β terms are obtained throughempirical calibration. We calibrated IGVf for rigidgrain-rich rocks by using 218 well to moderatelysorted samples with less than 10% matrix and lessthan 10% nonquartz cements. The samples are fromthe Norwegian continental shelf, are of Jurassic andCretaceous age, and were analyzed by us, as well as by Ehrenberg (1990, 1991, 1993) and S. N.Ehrenberg (1995, personal communication). Thesamples show present-day IGV values of 28.0 ±5.6%and show no significant variation with estimatedmaximum effective stress values over a range of from

Lander and Walderhaug 439

around 25 to 55 MPa. This lack of variation suggeststhat the samples are at or near their minimum com-pacted IGV values. As a result, the values can beused to define the IGVf term in the compactionfunction. The values are in good agreement withpreviously reported values for rigid grain-rich sand-stones from other basins (Paxton et al., 1990;Ajdukiewicz et al., 1991; Szabo and Paxton, 1991).

The β term is best constrained using samples thathave not achieved stable packing configurations. Tocalibrate β for rigid grain-rich rocks we used the dataof McBride et al. (1990) for Texas Eocene sandstonesthat have been exposed to maximum effective stress-es ranging from about 10 to 50 MPa. A value of 0.06MPa–1 provides a good correspondence betweenmodel predictions and measurements (Figure 2).

The most thoroughly documented study of theeffects of lithic fragments on compaction has beenmade by Pittman and Larese (1991). They conduct-ed an extensive set of compaction experimentsusing quartz grains mixed in various proportionswith volcanic, sedimentary, and metamorphic rockfragments. Correspondence between IGV predic-tions and measurements for the data set are shownas a function of effective stress in Figure 3, and theassociated calibrated parameters are listed in Table3. Over 85% of the model IGV predictions occurwithin the theoretical 95% confidence interval forpoint-count analysis. Virtually all of the predictionsthat are found outside of the confidence intervalcorrespond to experiments conducted at low effec-tive stress values where the experiment resultsshould be expected to diverge most strongly fromnatural sandstone behavior. In their experiments,Pittman and Larese (1991) used a Vibratool™ to

initiate grain rearrangement before compressingthe samples, resulting in IGV values of 32.2 ±3.7%.The Exemplar calibrations, however, assume initialporosities of 38% to simulate compaction of naturalsamples more realistically.

440 Predicting Porosity

Figure 1—Causes of surface area reduction during sandstone diagenesis.

IGV (%)

Max

imu

m B

uri

al D

epth

(m

)(H

ydro

stat

ic P

ress

ure)

0 10 20 30 40 500

1000

2000

3000

4000

5000

Max

imu

m E

ffec

tive

Str

ess

(MP

a)

0

10

20

30

40

50

60

Figure 2—Calibration of the Exemplar compactionmodel for well sorted to moderately well sorted, rigidgrain-rich sandstones. The petrographic data are fromMcBride et al. (1990). IGV = intergranular volume.

Unfortunately, no other ductile grain-rich datasets were available to test the calibrated com-paction functions. Although the experimental com-paction trends provide useful insights into thebehavior of ductile grain-rich sandstones, themodel IGV decline functions for ductile grain-richsandstones probably should be considered to rep-resent the maximum likely IGV end members.Greater extents of compaction may occur givenmore realistic burial rates (Kurkjy, 1988) and can bedefined in the model by decreasing the IGVf term.

Note that all compaction data could be matchedusing a β value of 0.06 MPa–1 (with the exceptionof the highly ductile sandstone comprised of 75%weathered basalt grains). The apparent insensitivityof β to sandstone composition suggests that rates ofstress increase are generally of second-order impor-tance in controlling the extent of compaction. Bycontrast, the IGVf term is highly variable dependingon the mechanical strength of the sandstone. Theseresults indicate that whereas β can be assumed tobe a constant for all but the most ductile of sand-stones, IGVf must be empirically calibrated for spe-cific sandstone compositional and textural types.

Quartz Precipitation Kinetics

Using a and b parameters calibrated to Jurassicsandstones of the Norwegian shelf, we testedquartz cement predictions for two data sets thatrepresent end members in time and temperaturehistories. Cambrian quartzarenites from the Balticshield represent a low-heating-rate end memberwith maximum burial temperatures of approxi-mately 90°C. The second test data set lies at theopposite end of the time and temperature spec-trum and is composed of Pliocene–Pleistocenesandstones from the Gulf of Mexico Basin thatreached temperatures of as much as 140°C.

Model predictions match measured values with-in measurement uncertainties (Figure 4), suggest-ing that the kinetic parameters derived from cali-bration to the Jurassic sandstones of the Norwegiancontinental shelf are applicable to sandstones withwide ranges in burial histories.

APPLICATION OF EXEMPLAR TO RESERVOIRQUALITY ANALYSIS

Present-day sandstone porosity is a function ofthe sandstone’s textural and compositional charac-teristics and burial history. In this section we illus-trate how Exemplar can be used to assess theeffects of thermal history, timing of fluid overpres-sure development, grain size, and grain coating onreservoir quality. In addition, we show how themodel can provide probabilistic predictions ofreservoir quality prior to drilling when used in con-cert with Monte Carlo techniques.

Effect of Burial History on Porosity Loss

To illustrate the effects of burial history onmodel porosity predictions, we selected threequartzarenite sandstones that show a wide range ineffective stress and temperature histories (Figure5), but similar initial compositions and textures.The examples include (1) a Miocene sample fromthe Gulf of Mexico that was subject to rapid burialand early f luid overpressure development, (2) aJurassic sample from the North Sea that developedfluid overpressures late in its burial history and hasbeen exposed to high thermal gradients, and (3) aCambrian sample from the Baltic region that expe-rienced uplift and subsequent burial. Input for thesimulations was derived from the petrographic dataof Freeman (1990), Walderhaug (1994), and S.Bloch (1993, personal communication), and theburial histories of Shaw and Lander (1994),Walderhaug (1994), and Brangulis et al. (1993).Parameters for the compaction and quartz cemen-tation models are identical for all three simulations.

Despite similar initial texture and composition,model results for the three sandstones indicate sub-stantial differences in the relation of porosity topaleoburial depth, and show a much more com-plex porosity decline function than that ordinarilypredicted by porosity models (Figure 6A). Modelresults show that essentially all porosity loss is dueto compaction for samples at paleoburial depths ofless than 1500 m, as shown by the IGV/depth

Lander and Walderhaug 441

Table 3. Calibrated Compaction Parameters

Rigid Weathered Basalt Shale SlateGrains 25% 50% 75% 25% 50% 75% 25% 50% 75%

IGVf * 28 22 6 0 22 6 0 24 19 13β** 0.06 0.06 0.06 0.15 0.06 0.06 0.06 0.06 0.06 0.06

*IGVf = stable packing intergranular volume (%).**β = exponential rate of intergranular volume loss with effective stress (1/MPa).

(Figure 6B) and bulk rock volume/depth plots(Figure 6C). The extent of shallow compaction issimilar for the Baltic and North Sea examplesbecause both are assumed to have been underhydrostatic fluid pressures over this interval. Bycontrast, the early and extensive f luid overpres-sures experienced by the Gulf of Mexico examplesignificantly reduced the extent of compaction(Figure 6B). Although the North Sea example alsopresently occurs within a zone of extensive fluidoverpressure, the development of the overpressureis assumed to have occurred after the sandstonehad been buried to approximately 2500 m (Figure5C). Simulations suggest that such deep overpres-sure development has little effect on porositybecause compaction essentially is complete by this

442 Predicting Porosity

Weathered Basalt0

10

20

30

40

50

0 10 20 30 40IGV (%)

Max

imum

Eff

ectiv

e S

tres

s (M

Pa)

Shale0

10

20

30

40

50

0 10 20 30 40IGV (%)

Max

imum

Eff

ectiv

e S

tres

s (M

Pa)

Slate0

10

20

30

40

50

0 10 20 30 40IGV (%)

Max

imum

Eff

ectiv

e S

tres

s (M

Pa) 25% Ductile Grains

50% Ductile Grains75% Ductile Grains

25% Exemplar50% Exemplar75% Exemplar

0

5

10

15

20

25

30

35

0 5 10 15 20 25 30 35Thin Section Quartz Cement (%)

Pre

dic

ted

Qu

artz

Cem

ent

(%)

Baltic, Cambrian

Gulf of Mexico, Pliocene–Pleistocene

58 samples

Figure 3—Calibration of the compaction model to sand-stones with a variety of types and abundances of ductilegrains. The experimental data, from Pittman and Larese(1991), are shown by the symbols, and the calibratedmodel results are shown by the lines. IGV = intergranu-lar volume.

Figure 4—Comparison of predicted and measuredquartz cement abundances for samples representingend members in heating rates using quartz cementationkinetics calibrated to Jurassic samples from the Norwe-gian continental shelf. The model provides accurate pre-dictions for shallow Cambrian samples from the Balticregion [petrographic data from S. Bloch (1993, personalcommunication); burial history data from Brangulis etal. (1993)], as well as for deeply buried Pliocene–Pleistocene samples from the Gulf of Mexico Basin [pet-rographic data of Milliken (1985)].

depth and thus is irreversible. Note that althoughthe high fluid overpressures experienced by theGulf of Mexico example have reduced the extent ofcompaction, the sample experienced rates of com-paction that are many times that of the North Seaand Baltic samples (Figure 6E). High rates of stressincrease may explain the common occurrence offractured grains in these rocks (Freeman, 1990).

Although the North Sea and Baltic examplesshow that elapsed time plays no role in simulatedporosity loss due to compaction (except wheneffective stress varies), time is a critical controlon the extent of quartz cementation when paleo-temperatures exceed 70°C (approximately 1500 mburial). The effect of time is particularly wellillustrated by the Baltic example by comparingpredicted quartz cement fractions at 2200 m ofburial (Figure 6D) as a function of the sandstone’sburial history (Figure 5A). The modeled quartzcement fraction increases from 4% during the ini-tial burial phase to 2200 m to 15% after the 260-m.y. period during which uplift and renewedsubsidence returned the sample to 2200 m. Bycontrast, models that relate porosity to thermalindicators such as vitrinite reflectance would pre-dict that porosity would remain essentially con-stant over this interval.

The model suggests that small, but significant,volumes of quartz cement can precipitate at com-paratively shallow depths given sufficient time.This is shown by the Baltic sandstone, which ismodeled to have about 2% quartz cement at apaleoburial depth of 1750 m. Small amounts ofquartz cement at such shallow depths could act tosignificantly strengthen the rock framework, there-by retarding compaction. Such a scenario is consis-tent with the anomalously high IGV values dis-played by the Baltic data set (34 ±6%) (S. Bloch,1993, personal communication).

Temperature also plays a critical role in control-ling quartz cementation, as is illustrated by themodel behavior for North Sea and Baltic samples.Higher temperatures associated with the high ther-mal gradients and greater burial of the North Seaexample resulted in net rates of quartz cementa-tion that are over an order of magnitude greaterthan the greatest displayed by the Baltic sample(Figure 6F).

Effect of Grain Coating on Porosity Loss

Grain coatings have long been recognized as acritical control on quartz cementation in sandstonesbecause they can prevent or impede the nucleationof syntaxial quartz overgrowths on detrital quartzgrains [see review by Ehrenberg (1993)]. Coatingscommonly are made up of authigenic chlorite

(Ehrenberg, 1993), microcrystalline quartz (Rammand Forsberg, 1991; Ramm, 1994; Aase et al., 1996),siderite (R. E. Larese, 1997, personal communica-tion), and detrital clay and Fe-oxyhydroxides.

Lander and Walderhaug 443

0

1000

2000

3000

4000

5000

60000100200300400500600

(A)

Baltic (Cambrian)Norwegian Shelf (Jurassic)

Time (Ma)

Gulf of Mexico (Miocene)

0

50

100

150

2000100200300400500600

(B)

Time (Ma)

0

5

10

15

20

25

30

35

400100200300400500600

(C)

Time (Ma)

Bu

rial

Dep

th (

m)

Tem

per

atu

re (

°C)

Eff

ecti

ve S

tres

s (M

Pa)

Figure 5—Burial history input data used for comparisonin modeled compaction and quartz cementation as afunction of burial history for the Baltic, the Norwegianshelf, and the Gulf of Mexico Basin. (A) Burial depth his-tory. (B) Temperature history. (C) Effective stress histo-ry. The data used as input for the simulations werederived from the burial histories of Brangulis et al.(1993), Shaw and Lander (1994), and Walderhaug (1994).

0 10 20 30 40

0

1000

2000

3000

4000

5000

6000

Porosity (%)

(A)Baltic (Cambrian)Norwegian Shelf (Jurassic)Gulf of Mexico (Miocene)Measurements

0 10 20 30 40

0

1000

2000

3000

4000

5000

6000

IGV (%)

(B)

0 0.2 0.4 0.6 0.8 1

0

1000

2000

3000

4000

5000

6000

Bulk Rock Volume (cm3)

(C)

0 5 10 15 20 25

0

1000

2000

3000

4000

5000

6000

Quartz Cement (%)

(D)

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

0

1000

2000

3000

4000

5000

6000

Rate of Porosity Loss (cm3/m.y.)

From Compaction

(E)

0 0.002 0.004 0.006 0.008 0.01

0

1000

2000

3000

4000

5000

6000

Rate of Porosity Loss (cm3/m.y.)

From Quartz Cementation

(F)

Pal

eob

uri

al D

epth

(m

)

Pal

eob

uri

al D

epth

(m

)

Pal

eob

uri

al D

epth

(m

)

Pal

eob

uri

al D

epth

(m

)

Pal

eob

uri

al D

epth

(m

)

Pal

eob

uri

al D

epth

(m

)

Perhaps the most conclusive study to date relatingthe textural effect of grain coating chlorite onquartz cementation is that of Byrnes and Wilson(1994) on the St. Peter and Mt. Simon sandstones ofthe Illinois basin. These workers showed thatalthough there is no simple relationship betweenthe overall abundance of chlorite with quartzcement, there is a systematic trend in quartzcement abundance with the visually estimatedcompleteness of chlorite coating on quartz grains(Figure 7).

The relationship between grain coating andquartz cement abundance is well matched byExemplar predictions based on the petrographicdata of Byrnes and Wilson (1994) and the burialhistory reconstruction of Bethke et al. (1991)(Figure 7). Model predictions for the average com-position and grain size come close to matching themedian measured values. In addition, much of thevariation in quartz cement abundance for a givengrain coating can be accounted for by grain sizevariations, as illustrated by the model results forgrain sizes one standard deviation toward coarserand finer textures as reported by Byrnes andWilson (1994). An apparent discrepancy betweenthe measurements and the model occurs at 100%grain coating, where the model predicts no quartz

Figure 6—Comparison of modeled compaction and quartz cementation as a function of burial history for sampleswith widely varying burial histories (shown in Figure 5), but similar initial textures and compositions. (A) Porositywith paleoburial depth. (B) IGV (intergranular volume) with paleoburial depth. (C) Bulk rock volume with paleo-burial depth. (D) Quartz cement abundance with paleoburial depth. (E) Rate of porosity loss due to compactionwith paleoburial depth. (F) Rate of porosity loss due to quartz cementation with paleoburial depth. Measured dataare shown with the solid squares on (A), (B), and (D).

Figure 7—The completeness of grain coatings andquartz cement abundance for St. Peter and Mt. Simonsandstones from the Illinois basin. The measured dataare from Byrnes and Wilson (1994), and the curves aremodel results using petrographic data of Byrnes andWilson (1994) and burial history results of Bethke et al.(1991) as input.

0

4

8

12

16

20

0Q

uar

tz C

emen

t (%

Vo

lum

e)20 40 60 80

Clay Coating on Quartz Grains (% Area)100

Byrnes and Wilson (1994)

Average

1σ Fine

1σ Coarse

Table 4. Input Uncertainties Used for Monte Carlo Example

Distribution Reservoir 1 Reservoir 2

CompositionInitial porosity Normal 45 ±3% 45 ±3%Initial quartz grain Normal 44 ±5% 36 ±5%Initial matrix Normal 1.5 ±2.0% 1.5 ±2.0%Grain size Normal 0.25 ±0.07 mm 0.21 ±0.05 mmGrain coating Normal 2 ±0% 2 ±0%

Burial History*Temperature 21 Ma Triangular 80–115°C 100–150°CMbd 21 Ma Triangular 1700–2570 m 2500–3800 mTemperature 64 Ma Triangular 75–110°C 105–140°CMbd 64 Ma Triangular 1500–1925 m 2400–3000 mTemperature 89 Ma Normal 102 ±10°C 130 ±10°CMbd 89 Ma Normal 1940 ±200 m 2910 ±200 m

Model AlgorithmsQuartz kinetics “b” Normal 0.022 ±0.001 mol/cm2s 0.022 ±0.001 mol/cm2sMinimum temperature for

quartz cementation Normal 70 ±7°C 70 ±7°Cβ∗∗ Log normal 0.06 ±0.01 MPa–1 0.06 ±0.01 MPa–1

IGVƒ† Normal 27 ±2% 22 ±3%

*Burial depth and temperature uncertainties are defined with correlation coefficients of 0.9. Mbd = maximum burial depth.**β = exponential rate of intergranular volume loss with effective stress (1/MPa).†IGVƒ = stable packing intergranular volume.

cement, whereas measurements indicate as much as2% cement (Figure 7). Model results consistent withthe median value of 1.2% quartz cement suggestthat, on average, approximately 3% of the quartzgrain surfaces are uncoated. This value, in fact, isconsistent with the 10% gap between the 90 and100% categories used by Byrnes and Wilson (1994)in their visual estimations. Thus, the model appearsto be robust given the uncertainties in the individualsample grain sizes, the measurement technique forgrain coating, and the burial histories of the samples.

A Stochastic Approach to Reservoir QualityPrediction

Monte Carlo techniques provide a means ofmore rigorously assessing the probable porosity fora prospective reservoir by incorporating uncertain-ties and variabilities in model input data. In addi-tion, when coupled with sensitivity analysis, MonteCarlo methods can be a useful aid in the quest toreduce predictive uncertainty because they helpidentify the input parameters that cause the great-est variability in the model predictions.

Uncertainties in the sandstone depositional tex-ture and composition are best obtained throughanalysis of petrographic data and depositional mod-els for mature basin areas, or analog settings forfrontier regions. Uncertainties in burial history dataare derived through basin modeling sensitivity stud-ies. To obtain uncertainties in the input parametersfor the compaction and quartz cementation algo-rithms, we use optimization techniques to find val-ues that match thin section point-count analysesgiven the “most likely” burial history input. Thecollective distribution of values obtained throughoptimization of a suite of petrographically analyzedsamples can then be used to define the uncertain-ties in the parameters used in the compaction andquartz cementation algorithms.

Once the input parameter uncertainties aredefined, it is a relatively simple matter to conduct aMonte Carlo simulation. An arbitrary number ofindividual simulations are run (typically numberingin the thousands) to obtain a frequency distributionfor the output parameter of interest. As an exam-ple, we have taken an undrilled prospect with twopotential reservoir intervals. The most significant ofthe input uncertainties are summarized in Table 4.

446 Predicting Porosity

Figure 8—Probable distribution inporosity given uncertainties inmodel input parameters for twopotential reservoir intervals in aprospect from a frontier setting.Each Monte Carlo simulationincorporates 2000 realizations.

0 10 205 15 250

0.008

0.015

0.023

0.030

Pro

bab

ility

Predicted Present-Day Porosity (%)

Reservoir 1

0 10 205 15 250

0.037

0.074

0.110

0.147

Pro

bab

ility

Predicted Present-Day Porosity (%)

Reservoir 2

Uncertainties in the burial history were derivedfrom basin modeling sensitivity studies, anduncertainties in the initial sandstone textures andcompositions are based on analog data from near-by subbasins, together with regional depositionalmodels. Uncertainties in model parameters arebased on optimization studies in other regions.Simulations suggest probable porosity values of14.9 ±4.4% and 3.1 ±2.6% for reservoirs 1 and 2,respectively (Figure 8). The simulations alsoshow that whereas 85% of the reservoir 1 simula-tions have greater porosity than the economiccutoff of 10%, only 0.2% of the reservoir 2 simula-tions have greater porosity than the economiccutoff despite their occurrence at the same welllocation. We also ranked the input parameterswith respect to their effect on the variability inthe present-day porosity prediction by determin-ing the contribution to variance (Table 5). Theanalysis indicates that whereas the greatest con-tribution to predicted variations in porosity forreservoir 1 is associated with uncertainties in theburial history, reservoir 2 is most sensitive touncertainty in grain size and the kinetic parame-ters for quartz cementation.

CONCLUSIONS

Exemplar provides useful insights into the evo-lution of sandstone porosity through geologictime. The model accuracy is reasonably good inview of the uncertainties in both calibration andinput data for the test data sets, and in that itreproduces measured IGV and quartz cementfrom samples exposed to a wide range of burialhistories. Exemplar porosity predictions can be

made in probabilistic terms through the applica-tion of parameter optimization and Monte Carlotechniques.

REFERENCES CITEDAase, N. E., P. A. Bjørkum, and P. H. Nadeau, 1996, The effect of

grain-coating microquartz on preservation of reservoirporosity: AAPG Bulletin, v. 80, p. 1654–1673.

Ajdukiewicz, J. M., S. T. Paxton, and J. O. Szabo, 1991, Deepporosity preservation in the Norphlet Formation, Mobile Bay,Alabama (abs.): AAPG Bulletin, v. 75, p. 533.

Athy, L. F., 1930, Density, porosity, and compaction of sedimentaryrocks: AAPG Bulletin, v. 14, p. 1–24.

Atkins, J. E., and E. F. McBride, 1992, Porosity and packing ofHolocene river, dune, and beach sands: AAPG Bulletin, v. 76,p. 339–355.

Awwiller, D. N., and L. L. Summa, 1997, Quartz cement volumeconstraints on burial history analysis: an example from theEocene of western Venezuela (abs.): AAPG Annual ConventionProgram, p. A6.

Beard, D. C., and P. K. Weyl, 1973, Influence of texture onporosity and permeability of unconsolidated sand: AAPGBulletin, v. 57, p. 349–369.

Bethke, C. M., J. D. Reed, and D. F. Oltz, 1991, Long-rangepetroleum migration in the Illinois basin: AAPG Bulletin, v. 75,p. 925–945.

Bjørkum, P. A., 1994, How important is pressure in causingdissolution of quartz? (abs.): AAPG Annual ConventionProgram and Abstracts, v. 3, p. 105.

Bloch, S., 1991, Empirical prediction of porosity and permeabilityin sandstones: AAPG Bulletin, v. 75, p. 1157–1160.

Bloch, S., 1994a, Importance of reservoir prediction inexploration, in M. D. Wilson, ed., Reservoir quality assessmentand prediction in clastic rocks: SEPM Short Course 30, p. 5–8.

Bloch, S., 1994b, Secondary porosity in sandstones: significance,origin, relationship to subaerial unconformities, and effect onpredrill reservoir quality prediction, in M. D. Wilson, ed.,Reservoir quality assessment and prediction in clastic rocks:SEPM Short Course 30, p. 137–160.

Bloch, S., 1994c, Case history—offshore mid-Norway/TaranakiBasin, New Zealand/San Emigdio area, California, in M. D.Wilson, ed., Reservoir quality assessment and prediction inclastic rocks: SEPM Short Course 30, p. 357–366.

Bloch, S., and K. P. Helmold, 1994, Approaches to predictingreservoir quality in sandstones: AAPG Bulletin, v. 79, p. 97–115.

Brangulis, A. P., S. V. Kanev, L. S. Margulis, and R. A. Pomerantseva,1993, in J. R. Parker, ed., Geology and hydrocarbon prospectsof the Paleozoic of the Baltic region: Petroleum Geology ofNorthwest Europe, Proceedings of the 4th Conference: TheGeological Society, London, p. 651–656.

Byrnes, A. P., and M. D. Wilson, 1994, Chapter 22. Case History—St. Peter and Mt. Simon Sandstones, Illinois Basin, in M. D.Wilson, ed., Reservoir quality assessment and prediction inclastic rocks: SEPM Short Course 30, p. 385–394.

Clarke, R. H., 1979, Reservoir properties of conglomerates andconglomeratic sandstones: AAPG Bulletin, v. 63, p. 799–809.

Dickinson, W. W., and J. D. Ward, 1994, Low depositionalporosity in eolian sands and sandstones, Namib Desert: Journalof Sedimentary Research, v. A64, p. 226–232.

Dunn, T. L., 1994, Recognizing tectonic and compaction drivenquartz grain fracturing and annealing in the AlmondFormation, Green River basin, Wyoming (abs.): AAPG AnnualConvention Program and Abstracts, v. 3, p. 140.

Ehrenberg, S. N., 1989, Assessing the relative importance ofcompaction processes and cementation to reduction ofporosity evolution of Pliocene sandstones, Ventura basin,California: Discussion: AAPG Bulletin, v. 73, p. 1274–1276.

Lander and Walderhaug 447

Table 5. Sensitivity Results from Monte Carlo Exampleas Indicated by Contribution to Variance*

Reservoir 1 Reservoir 2

CompositionInital quartz grain 4.5 10.8Initial matrix 6.0 2.8Grain size (mm) 17.9 33.0

Burial HistoryMbd** and °C, 21 Ma 22.0 10.6Mbd and °C, 64 Ma 10.2 7.3Mbd and °C, 89 Ma 14.9 4.7

Model AlgorithmsQuartz kinetics “b” 16.9 26.7Minimum IGV† 6.7 3.7

Sum 99.1 99.6

*All units are in % of contribution to variance.**Mbd = maximum burial depth.†IGV = intergranular volume.

Ehrenberg, S. N., 1990, Relationship between diagenesis andreservoir quality in sandstones of the Garn Formation,Haltenbanken, mid-Norwegian continental shelf: AAPGBulletin, v. 74, p. 1538–1558.

Ehrenberg, S. N., 1991, Kaolinized, potassium-leached zones atthe contacts of the Garn Formation, Haltenbanken, mid-Norwegian continental shelf: Marine and Petroleum Geology,v. 8, p. 250–269.

Ehrenberg, S. N., 1993, Preservation of anomalously high porosityin deeply buried sandstone by grain-coating chlorite: examplesfrom the mid-Norwegian continental shelf: AAPG Bulletin, v. 77, p. 1260–1286.

Freeman, C. W., 1990, Diagenesis of Miocene sandstones andshales, southern Louisiana Gulf Coast: Master’s thesis,University of Missouri, Columbia, Missouri, 94 p.

Füchtbauer, H., 1967, Influences of different types of diagenesison sandstone porosity: Proceedings of 7th World PetroleumCongress, v. 2, p. 353–369.

Giles, M. R., and R. B. de Boer, 1990, Origin and significance ofredistributional secondary porosity: Marine and PetroleumGeology, v. 6, p. 261–269.

Heald, M. T., 1956, Cementation of Simpson and St. Petersandstones in parts of Oklahoma, Arkansas, and Missouri:Journal of Geology, v. 58, p. 624–633.

Houseknecht, D. W., 1988, Intergranular pressure solution in fourquartzose sandstones: Journal of Sedimentary Petrology, v. 58,p. 228–246.

Kurkjy, K. A., 1988, Experimental compaction studies of lithicsands: Master’s thesis, University of Miami, Miami, Florida, 101 p.

Land, L. S., and S. P. Dutton, 1978, Cementation of a Pennsylvaniandeltaic sandstone: isotopic data: Journal of SedimentaryPetrology, v. 48, p. 1167–1176.

Lander, R. H., V. Felt, L. Bonnell, and O. Walderhaug, 1997a,Utility of sandstone diagenetic modeling for basin historyassessment (abs.): AAPG Annual Convention Program, p. A66.

Lander, R. H., O. Walderhaug, and L. Bonnell, 1997b, Applicationof sandstone diagenetic modeling to reservoir qualityprediction and basin history assessment: Memorias del ICongreso Latinoamericano de Sedimentología, Venezolana deGeólogos Tomo I, p. 373–386.

Laubach, S. E., 1997, A method to detect natural fracture strike insandstones: AAPG Bulletin, v. 81, p. 604–623.

Lundegard, P. D., 1991, Sandstone porosity loss—a “big picture”view of the importance of compaction: Journal of SedimentaryPetrology, v. 62, p. 250–260.

McBride, E. F., T. N. Diggs, and J. C. Wilson, 1990, Compaction ofWilcox and Carrizo sandstones (Paleocene–Eocene) to 4420 m,Texas Gulf Coast: Journal of Sedimentary Petrology, v. 61, p. 73–85.

Meshri, I. D., 1990, An overview of chemical models and theirrelationship to porosity prediction in the subsurface, in I. D.Meshri and P. J. Ortoleva, eds., Prediction of reservoir qualitythrough chemical modeling: AAPG Memoir 49, p. 45–53.

Milliken, K. L., 1985, Petrology and burial diagenesis ofPlio–Pleistocene sediments, northern Gulf of Mexico: Ph.D.dissertation, University of Texas, Austin, Texas, 112 p.

Palmer, S. N., and M. E. Barton, 1987, Porosity reduction,microfabric and resultant lithification in UK uncementedsands, in J. D. Marshall, ed., Diagenesis of sedimentarysequences: Geological Society Special Publication 36, p. 29–40.

Paxton, S. T., J. O. Szabo, C. S. Calvert, and J. M. Ajdukiewicz, 1990,Preservation of primary porosity in deeply buried sandstones: anew play concept from the Cretaceous Tuscaloosa Sandstone ofLouisiana (abs.): AAPG Bulletin, v. 74, p. 737.

Pittman, E. D., 1979, Porosity, diagenesis and productivecapability of sandstone reservoirs: SEPM Special Publication26, p. 159–173.

Pittman, E. D., and R. E. Larese, 1991, Compaction of lithic sands:experimental results and applications: AAPG Bulletin, v. 75, p. 1279–1299.

Ramm, M., 1994, Porosity depth trends in Upper Jurassicreservoirs, Norwegian central graben: an example of porositypreservation at deep burial by grain-coating microquartz (abs):AAPG Annual Convention Program and Abstracts, p. 241.

Ramm, M., and A. W. Forsberg, 1991, Porosity vs. depth trends inUpper Jurassic sandstones from the Cod Terrace area, centralNorth Sea, in M. Ramm, ed., Porosity depth trends in reservoirsandstones: Ph.D. thesis, University of Oslo, Oslo, Norway,308 p.

Schneider, F., M. Bouteca, and G. Vasseur, 1994, Validity of theporosity/effective stress concept in sedimentary basinmodeling: First Break, v. 12, p. 321–326.

Shaw, C. A., and R. H. Lander, 1994, Testing the sensitivity ofhydrocarbon migration and overpressure development torock, fault, and fluid properties at the field scale (abs.): AAPGAnnual Convention Program and Abstracts, v. 3, p. 257.

Siever, R., and W. N. Stone, 1994, Quantitative petrologicconstraints on basin paleohydrologic models (abs.): AAPGAnnual Convention Program and Abstracts, p. 258–259.

Sippel, R. F., 1968, Sandstone petrology, evidence fromcathodoluminescence petrography: Journal of SedimentaryPetrology, v. 38, p. 530–554.

Suchecki, R. K., and S. Bloch, 1988, Complex quartz overgrowthsas revealed by microprobe cathodoluminescence (abs.): AAPGBulletin, v. 72, p. 252.

Sweeney, J. J., and A. K. Burnham, 1990, Evaluation of a simplemodel of vitrinite reflectance based on chemical kinetics:AAPG Bulletin, v. 74, p. 1559–1570.

Szabo, J. O., and S. T. Paxton, 1991, Intergranular volume (IGV)decline curves for evaluating and predicting compaction andporosity loss in sandstones (abs.): AAPG Bulletin, v. 75, p. 678.

Taylor, J. M., 1950, Pore space reduction in sandstones: AAPGBulletin, v. 34, p. 710–716.

Thomson, A., 1959, Pressure solution and porosity, in H. A.Ireland, ed., Silica in sediments: SEPM Special Publication 7, p. 92–111.

Ungerer, P., J. Burrus, B. Doligez, P. Y. Chenet, and F. Bessis,1990, Basin evaluation by integrated two-dimensionalmodeling of heat transfer, fluid flow, hydrocarbon generation,and migration: AAPG Bulletin, v. 74, p. 309–335.

Walderhaug, O., 1994, Precipitation rates for quartz cement insandstones determined by fluid-inclusion microthermometryand temperature-history modeling: Journal of SedimentaryResearch, v. A64, p. 324–333.

Walderhaug, O., 1996, Kinetic modeling of quartz cementationand porosity loss in deeply buried sandstone reservoirs: AAPGBulletin, v. 80, p. 731–745.

Walderhaug, O., R. H. Lander, P. A. Bjørkum, E. H. Oelkers, K. Bjørlykke, and P. H. Nadeau, in press, Modelling quartzcementation and porosity in reservoir sandstones—examplesfrom the Norwegian continental shelf, in R. Worden, ed.,Quartz cementation in oil field sandstones: InternationalAssociation of Sedimentologists Special Publication.

Waples, D. W., and H. Kamata, 1993, Modelling porosityreduction as a series of chemical and physical processes, inA. G. Dóre et al., eds., Basin modelling: advances andapplications: Norwegian Petroleum Society Special Publication3, p. 303–320.

Wilson, J. C., and E. F. McBride, 1988, Compaction and porosityevolution of Pliocene sandstones, Ventura basin, California:AAPG Bulletin, v. 72, p. 664–681.

Wilson, M. D., 1984, Clastic diagenesis, in Geology for engineers, shortcourse notes: The Petroleum Society of the Canadian Institute ofMining, Metallurgy and Petroleum, Calgary Section, 24 p.

Wilson, M. D., 1994, Introduction, in M. D. Wilson, ed., Reservoirquality assessment and prediction in clastic rocks: SEPM ShortCourse 30, p. 1–4.

Wilson, M. D., and P. T. Stanton, 1994, Diagenetic mechanisms ofporosity and permeability reduction and enhancement, inM. D. Wilson, ed., Reservoir quality assessment and prediction

448 Predicting Porosity

in clastic rocks: SEPM Short Course 30, p. 59–119.Wood, J. R., 1994, Geochemical models, in M. D. Wilson, ed.,

Reservoir quality assessment and prediction in clastic rocks:SEPM Short Course 30, p. 23–41.

Wood, J. R., and A. P. Byrnes, 1994, Alternate and emergingmethodologies in geochemical and empirical modeling, inM. D. Wilson, ed., Reservoir quality assessment and predictionin clastic rocks: SEPM Short Course 30, p. 395–400.

Lander and Walderhaug 449

Rob Lander

Rob Lander is a senior staff geolo-gist with Geologica in Stavanger,Norway, where he has been in-volved in the development andapplication of diagenetic and basinmodels since 1993, when he joinedGeologica’s parent company, Roga-land Research. He obtained a Ph.D.in 1991 from the University ofIllinois, where he studied the fluid-flow and geochemical controls onthe alteration of volcanogenic sediments. While at ExxonProduction Research from 1990 to 1993, he worked onsimulations of fluid overpressure development.

Olav Walderhaug

Olav Walderhaug is a staff geolo-gist at Statoil’s exploration technol-ogy division in Stavanger, Norway.He received his B.Sc. and M.Sc.degrees in petroleum geology fromthe University of Bergen, and aD.Sc. degree in sandstone diagene-sis from the University of Oslo. Hismain research interests are withinthe field of sandstone diagenesisand related topics, including devel-opment of quantitative predictive models for quartzcementation and porosity evolution in reservoir sand-stones.

ABOUT THE AUTHORS