* Currently with ExxonMobil Upstream Research Company

Copyright 2000, Society of Petroleum Engineers Inc.

This paper was prepared for presentation at the 2000 SPE Annual Technical Conference andExhibition held in Dallas, Texas, 14 October 2000.

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AbstractGeophysicists, geologists, and reservoir engineers can nowroutinely build reservoir geologic models with ten milliongeologic cells and more than one thousand geologic layers.This explosion in reservoir detail capability presents newchallenges for existing upscaling methods. Uplayering (thefirst step of upscaling) is a technique that provides reservoirengineers with optimal geologic layer-grouping schemes forsimulation model construction. Much uplayering is still doneby hand by reservoir engineers. Even though some advancedmethods can provide automatic tools for uplayering, they arelimited in the applicable model size and are alsocomputationally expensive. This paper presents a practical andefficient method for uplayering of multimillion-cell geologicmodels. The proposed method defines a displacing frontconductivity, a combination of porosity, permeability, andfacies (in terms of relative permeability, endpoint saturation,and various facies rules), as the uplayering property. Use ofthe new property ensures that the most important geologicfeatures for fluid flow simulation can be preserved afteruplayering. The new method utilizes a residual optimizationtechnique to determine the optimal geologic layer-groupingscenario for a given number of simulation layers. The methodis so efficient that multiple optimal grouping scenarios, fromthe one simulation layer model to the model consisting of allgeologic layers, can be generated in a short time, allowing theinspection of all possible combinations of layer-groupingscenarios. A residual curve (the difference of the definedproperty between the fine-layer and coarse-layer models) isproduced from exhaustive analysis of all possible layeringcombinations. Using the residual curve, engineers are able todetermine the number of simulation layers needed based ontheir tolerance of possible loss of fine-layer geologic features.The uplayering method has been successfully employed inmost of recent major reservoir study projects in Mobil and

results from three of these studies will be included in thispaper.

IntroductionSimulating a petroleum reservoir directly using multimillion-cell to multibillion-cell geologic models challenges reservoirmodeling resources, on both hardware and software. Due toparallel computing techniques, the most advanced reservoirsimulators (existing or under development) in the petroleumindustry now can handle up to several million cells by usingmore than one hundred CPUs. Theoretically, one can simulatemultimillion to multibillion cells if enough (hundreds tothousands) CPUs are dedicated to reservoir simulation.Unfortunately, reservoir engineers may always findthemselves one step behind geologic modeling techniques interms of the size of a model. This is due, in part, to the factthat geologic models are constructed using static, algebraicsystems while reservoir simulation systems are dynamicsolutions of partial differential equations. While reservoirengineers are trying hard to simulate multimillion cell models,geologists are building multibillion cell geologic models. Themajor challenge for reservoir flow modelers is to simulate areservoir efficiently without a significant loss of reservoirheterogeneities and geologic features. Upscaling provides asolution for this challenge.

Construction of reservoir simulation layers based on the givengeologic layers, i.e. vertical gridding, is the first step ofupscaling. Traditionally, there are three ways for reservoirengineers to construct their simulation layers. (1) Use geologiclayers directly when the number of geologic layers (nz) isequal to the number of simulation layers (Nz). (2) Groupgeologic layers when nz is larger than Nz, and (3) split some ofthe geologic layers when nz is smaller than Nz. Sincetraditional simulation layer construction is a manual process,reservoir engineers are only comfortable with small geologicmodels (less than ~50 layers per model). For a large geologicmodel, with several hundred to thousands of geologic layers,reservoir engineers have no choice but to uniformly groupgeologic layers to form simulation layers. As a result, criticalreservoir heterogeneities in the geologic model built bygeologists may be lost. Furthermore, this over-homogenizedmodel may cause large errors in reservoir history match andperformance prediction and may result in bad businessdecisions. Since most geologic models today and in the futurewill contain more layers than simulation models, the

SPE 62927

Optimal Uplayering for Scaleup of Multimillion-Cell Geologic ModelsD. Li and B. Beckner, Mobil Technology Company*

2 D. LI AND B. BECKNER SPE 62927

construction of simulation layers in this paper only refers togeologic-layer grouping or uplayering. A good uplayeringmethod should provide reservoir engineers with the bestgeologic-layer grouping schemes and the number ofsimulation layers needed based on their tolerance for the lossof the fine-layer geologic features for simulation.About 35 years ago, Testerman1 published a statisticalreservoir-zonation method for reservoir engineers to identifyreservoir zones from detailed permeability data along wellbores. The reservoir zones identified have a minimumvariation of permeability internally and a maximum variationbetween zones. Unfortunately, Testermans idea was ignoreduntil 1995 when Li et al.2 extended his idea to globalupscaling, i.e. use of three-dimensional unequal spaced gridcells to capture reservoir heterogeneities residing in geologicmodels. The Li et al.2 method provides simulation modelconstruction with the best geologic-layer-grouping schemebased on permeability variation for a given number ofsimulation layers. The major advantage of this method is thatit is relatively fast, compared to existing methods3-6, becauseno numerical simulation is used in the uplayering process.Stern and Dawson7 recently presented a fast method that issimilar to Li et al.s work with two major improvements: afaster search algorithm with better defined optimizationproperties (breakthrough time and flux) and a way todetermine the number of simulation layers needed. Other non-simulation-based methods can be found in commercialsoftware packages8,9. Contrasted to the non-simulation-basedmethods is the group of simulation-based methods6-12. Sincesimulation-based methods need to solve the single-phase flowequation to calculate flux distribution, they are limited tosmall geologic models (< 2 million cells). Anotherdisadvantage is that simulation-based methods do not give aguide as to the number of simulation layers needed for a givengeologic model as they are too computationally extensive togenerate multiple simulation layer model scenarios. Inaddition, all existing methods ignore geologic facies (rocktypes), the most important factor for geologic modeling andfluid flow simulation, during uplayering.

In this paper, we present a practical and efficient uplayeringmethod. The proposed method is facies-based, fast, and size-independent (>100 million cells). The new method provides ameasure of the difference between a geologic model and itsoffspring simulation models through a residual curve. Thiscurve is also a guide to reservoir engineers as to the number ofsimulation layers needed to maintain fidelity with the geologicmodel.

New MethodFor ease of presentation, the demonstration of the new methodis arranged in the following four sections: residualoptimization, uplayering property, residual curve, andextensions.

Residual OptimizationTo illustrate the proposed idea, let us look at how engineersgroup geologic layers for simulation models. Figure 1

illustrates layer-averaged permeability (vertical axis) as afunction of depth (horizontal axis) for a 100-layer geologicmodel. This is a synthetic model with 100 1-foot layers.Suppose it is decided that only 25 simulation layers can bemodeled, for reasons such as computer memory limitation.The engineers may use a traditional approach, i.e. groupingevery four geologic layers into one simulation layer, by which25 4-foot simulation layers are obtained. After an appropriateaverage permeability is assigned to the coarse simulationlayers, their simulation model is complete and ready forsimulation. Unfortunately this simulation model differssignificantly from the original geologic model. Figure 1 showsthat high permeability zones and low permeability barriershave been drastically homogenized or completely lost duringtraditional uplayering from the geologic model to thesimulation model. For example, geologic layer 39-40 (shale,zero permeability layers) has been replaced by simulationlayer 10 (a 38 md permeable layer). As a result, simulationlayer 10 creates a connection between the upper zones(geologic layer 1-38) and the bottom zones (geologic layer 41-100) that is not seen in the geologic model.Unlike the above uniform-lumping approach, the proposednew method focuses on preserving reservoir heterogeneitiesand geologic details via a non-uniform layer-groupingstrategy. The strategy determines the optimal layer-groupingscheme from all possible groupings for a given number ofsimulation layers through minimizing the difference, calledresidual (shaded area in Fig. 2), between a geologic model andits offspring simulation model. This residual optimizationmethod optimally retains high permeability streaks and flowbarriers, and therefore the loss of reservoir heterogeneity isminimized. Figure 2 illustrates the 25 simulation layersconstructed by the residual optimization method. Thethickness of a simulation layer created by residualoptimization changes from 2 feet to 8 feet, i.e. simulationlayers contain between 2 to 8 geologic layers.

The residual function is defined as:

= = =

=

z y xn

k

n

j

n

i zyx

fijk

c

ijk

nnn

ppR

1 1 1

2)((1)

where R is residual; cijkp is the coarse-layer property mapped(downscaled) at the location (i, j, k) of the fine-layer model;

fijkp is the fine-layer property at the location (i, j, k) of the

fine-layer model; nx and ny are numbers of cells in the x and ydirections of the fine-layer model, respectively; and nz isnumber of layers of the fine-layer model. As we will see in thenext section, property p in Eq. 1 does not necessarily meanpermeability. It can be any property that is a gooddiscriminator to characterize reservoir heterogeneity. Wetemporally skip the definition of property p and leave it for theuplayering property section.Theoretically, Eq. 1 provides a thorough definition of theresidual, but its use is not computationally efficient, especiallyfor multimillion-cell models. For instance, consider a geologic

SPE 62927 OPTIMAL UPLAYERING FOR SCALE-UP OF MULTIMILLION-CELL GEOLOGIC MODELS 3

model with 200 layers and 107 cells in total. If we want toconstruct an optimal 20-layer simulation model, we need torun about 1.61026 searching cases (see Appendix A fordetails) of all possible 20-layer simulation models.Considering that each case needs 5107 subtracting,multiplying, and dividing operations, the total operations forsearching an optimal layer scheme will be 91033. It is notpractical for current computers to do this large number ofoperations.

In order to make the layer optimization feasible twosimplifications are needed. We first simplify Eq. 1 from a cell-based calculation to a layer-based calculation. Thissimplification will reduce the number of operations by a factorof the number of cells per layer (e.g. 50000 for the aboveexample). The second simplification is to change from thethorough combination search to a sequential search (seefollowing descriptions). The sequential search cuts downdrastically the number of searching cases (from 1.61026 to19900 for the above example).The layer-based calculation uses several statistical parametersto represent a heterogeneous physical quantity for a layer.There are three orders of heterogeneity that can be used tostatistically represent reservoir heterogeneity for a givenquantity in the layer. The first order heterogeneity, the mostimportant one, is the mean of the quantity for that layer. Thevariance or standard deviation of the quantity for the layerconstitutes the second order heterogeneity. The third orderheterogeneity consists of the spatial correlation of the quantity(e.g. correlation lengths, orientation, and aspect ratios). Thefirst order heterogeneity captures heterogeneity on meanbetween layers, the second order heterogeneity describesheterogeneity within a layer, and the third order heterogeneitycharacterizes the trend and distribution of a quantity within alayer. Since extensive geostatistic work is needed forestimating the third order heterogeneity, we exclude thisheterogeneity in this uplayering optimization.

Considering the above simplification, Eq. 1 can be rewritten asfollows:

==

+

=

zz n

k z

fk

c

kn

k z

fk

c

k

nw

n

ppR

1

2

1

2 )()( (2)where p and are the mean and standard deviation of aquantity for a layer, respectively; w is a weight that variesfrom 0 to 1. The layer-based residual (Eq. 2) significantlyreduces computation effort from the cell-based residual (Eq. 1)and makes the uplayering optimization more attractive forpractical applications. Like Eq. 1, the coarse-layer properties( p and ) have to be downscaled to the fine-layer system forresidual calculation in Eq. 2. For a correct use of Eq. 2, p and should be transformed to normal distributions by using a pnormal transformation13 or a normal score transformation14.

As we have seen previously, it is not practical to visit allpossible layer-grouping cases for a given number of

simulation layers, so an efficient procedure is needed foroptimal case selection. A sequential search is a very efficientmethod that can meet those needs. There are three major stepsfor the sequential search: initialization, first search, anditeration. Initialization is a process that defines a starting pointfor searching the optimal case for a given number ofsimulation layers. The major concern in the initialization is tohonor the layer boundaries that are known to be important andthat must be preserved in uplayering. For example, if geologicevent (or sequence) boundaries in a geologic model built byusing geologic modeling packages, such as SGM and IRAP,are known, then those boundaries should be used to initializethe search process. In addition to geologic events orsequences, some other layer boundaries that are known to beimportant to fluid flow simulation and should also be addedinto the initialization. If there is no layer boundary withconcern, the initialized layer scheme will contain only onesimulation layer. The initial simulation-layer scheme isconstructed by using these selected boundaries. Theinitialization is finished after the residual defined by Eq. 2 forthe initial scheme has been calculated.

Starting from the initial simulation-layer scheme, the next stepis to search to the next boundary for the new simulation-layerscheme. A thorough search is carried out to visit all geologic-layer boundaries except those having been selected. Residualsfor each boundary visit are calculated and compared to eachother. After the search is done, the layer boundary that resultsin the smallest residual is determined and then the newsimulation-layer scheme is constructed using the new selectedboundary. The new scheme that contains one more simulationlayer than the previous scheme forms the first search result.The minimum residual for the new scheme is then kept forfuture use.

The iteration procedure basically repeats the first searchprocess until the number of simulation layers needed isreached. If we assume that the needed simulation layers are Nzand the initial simulation layers selected are Ni, the total

number of searching cases is equal to=

z

i

N

Nj j , significantlylower than the thorough searching case. The proposedsequential search is so efficient that the desired simulationlayers grouping from a very large geologic model, to sayaround 1000 layers, can be determined in less than one CPUhour using an ordinary computer (NT or UNIX).

Uplayering PropertyThe uplayering property is a property used to estimate theresidual or the accuracy of reservoir uplayering. An uplayeringproperty should be an attribute that best characterizes fluidflow in a petroleum reservoir. There is no unique way todefine an uplayering property. Traditionally geoscientists aremore interested in facies and porosity while engineers aremore interested in permeability. That is to say, reservoirengineers first consider permeability whereas geologists firstlook at facies and porosity when they group layers. Figure 3

4 D. LI AND B. BECKNER SPE 62927

shows a typical relationship among (horizontal) permeability,porosity, and facies. It is clear that permeability, porosity, andfacies are individually not adequate enough to be anuplayering property. Suppose some engineers use (horizontal)permeability as an uplayering property, no matter how hardthey try for the case shown in Fig. 3, they will always putapples and oranges together, i.e. different facies and porositytogether. Different facies have different relative permeabilitycurves, different ratios of vertical and horizontalpermeabilities, and different capillary pressure curves.Therefore it may not be a good idea to put them together.Similarly, if geologists use porosity or facies as an uplayeringproperty, they may put high permeability streaks and flowbarriers together, and changing reservoir model flowcharacteristics. If we can find a way to combine all of the threeparameters into one unified parameter as an uplayeringproperty, we may capture the whole picture of a geologicmodel for uplayering.

Based upon fluid flow through a layered porous media, adisplacing front conductivity (DFC) can be defined as follows(see Appendix B for details):

)( fh Sfk

DFC = (3)where hk is horizontal permeability; is porosity; )( fSf is displacing front velocity at Sf (displacing front saturation).DFC effectively combines permeability, porosity, and facies,making it a good uplayering candidate. )( fSf is defined asa water displacing front velocity for water injection but it canbe extended to a gas displacing front velocity for gas injection.

There is another way to define an uplayering property that isindependent of injection fluid types. This uplayering propertyis named as facies rules (FR) as follows:

rk

aFR h += (4)where a is a scaling factor; and r is a rule that discriminatesfacies. The parameter r is defined as a series of numericalnumbers by which the difference between two facies in termsof geology can be quantitative. The absolute value of r isarbitrary and not meaningful, but the relative values(differences) between different r values are meaningful. Inorder to define facies that can or cannot be grouped, twodegrees of the differences between r values need to bespecified. The first-degree difference defines the facies thatcannot be grouped while the second-degree differencedetermines those facies that can be grouped. The first-degreedifference should be much larger than the second-degreedifference depending on objectives and requirements. Forexample, if we define three poor sands as 1, 2, and 3, and fourgood sands as 101, 102, 103, and 104 then the first-degreedifference is 100 and the second-degree difference is 1. That isto say, poor sand can only be grouped with poor sand ratherthan good sand, and vice versa.

The relative importance of /hk in FR is controlled by thescaling factor a. To emphasize the importance of facies, a canbe defined as follows:

rk

ba

h

=max)/(

(5)

where b is a scaling number between 0 and 1; r is theaverage of the first-degree differences of r; max)/( hk is themaximum of /hk in the fine-layer geologic model. Use ofEq. 5 ensures that the first part in the right side of Eq. 4 issmaller than the average of the first-degree changes of r (thesecond part).

Residual CurveAfter a proper uplayering property has been chosen, a residualcurve as a function of the number of simulation layers from agiven initial number of layers to the number of geologic layerscan be generated. This curve shows all possible uplayeringscenarios for a given geologic model. Generating the entireresidual curve is equivalent to the search of the number ofsimulation layers that equals to the number of geologic layers,i.e. Nz = nz. This unique advantage makes the proposedmethod very efficient and capable of handling very largegeologic models that may contain more than a billion cellsand/or more than 10,000 layers using the current computingresources.

Figure 4 shows a typical residual curve plot in which thehorizontal axis represents number of simulation layers and thevertical axis the residual measure. Residual normallydecreases as number of simulation layers increases. Theresidual curve is bounded by its maximum at the minimumnumber of simulation layers (e.g. number of geologicsequences) and by its minimum at the maximum number ofsimulation layers (i.e. number of geologic layers). Themaximum residual depends on the uplayering property chosenand the minimum residual is always equal to zero. A residualcurve is characterized by three different regions denoted as A,B, C in Fig. 4. Region A describes the fastest drop of residual,region B delineates a relative slow drop of residual, and regionC shows zero residual. The boundary between regions A andB shows the minimum number of simulation layers needed tocapture the most important geologic heterogeneity in thegeologic model. And the boundary between regions B and Cgives the minimum number of simulation layers needed tomake the simulation model identical to the geologic model.

The residual curve can provide reservoir engineers withquantitative knowledge on the amount of the loss of reservoirheterogeneity regarding a selected uplayering scenario and canhelp them make critical decisions for their simulation modelconstruction. Once they have decided the number ofsimulation layers needed, they can easily achieve the desiredscenario from the database that contains all scenarios for thegiven geologic model. After they have obtained the selectedscenario, they can edit the layering scheme if they want to. For

SPE 62927 OPTIMAL UPLAYERING FOR SCALE-UP OF MULTIMILLION-CELL GEOLOGIC MODELS 5

example, they can split very thick layers and group very thinlayers in the selected scenario table before upscaling is carriedout. In addition, reservoir engineers are able to extract severalscenarios for their sensitivity study at any time they need.

ExtensionsThe proposed method can be easily extended to more generalvertical gridding for simulation model construction. The firstextension is to use a different number of simulation layers indifferent areas. For example, when some specific areas aremore important and more heterogeneous than others, moresimulation layers can be used in those areas. This can be doneby using more layers in the important areas with local gridrefinement (LGR) and less layers in the unimportant areaswith coarse areal grids. The second extension is to tune thesimulation layers selected to match geologic heterogeneityboundaries for those cases that heterogeneity boundaries donot align with layer boundaries. This process involves layertuning in each column (nxny columns in total). Layer tuningreduces the residual and as a result decreases the loss ofreservoir heterogeneity caused by uplayering.

Field ApplicationsThe proposed method has been applied to most of Mobil'srecent major reservoir studies worldwide. We will onlyoutline three applications (reservoirs A, B, and C) in thispaper.

Reservoir A is a sandstone, heavy oil reservoir in California,USA. The reservoir A geologic model was built using a SGMframework and (RC)2 geostatistic package for properties. Thegeologic model was described with 4197 geologic cells perlayer and 379 geologic layers. The model framework wasconstructed based on 5 geologic sequences that were split intogeologic layers proportionally. Each sequence contained arange of 61 to 98 geologic layers. Figure 5 shows the residualcurve of reservoir A model uplayering. The uplayeringproperty used in this work is porosity since porosity is themost reliable attribute for this model. According to thisresidual curve, the fastest residual-drop region endsapproximately at 29 simulation layers and the residual drops tozero when number of simulation layers reaches 320. Use of320 simulation layers should be adequate enough to capture allof geologic details residing in the 397-layer geologic model,i.e. reducing 57 layers should not cause any loss of reservoirdetail. Figure 5 also suggests that use of 29 simulation layersshould preserve the most important heterogeneity of thegeologic model. Figures 6 and 7 demonstrate comparisonbetween the original 379-layer geologic model and theuplayered 29-layer simulation model. The former comparesthe layer-averaged porosity of the simulation model to that ofthe geologic model, while the latter shows a 3D distributionsbetween the two model.

Reservoir B is a deepwater sandstone reservoir in Nigeria.Reservoir B geologic model was built using Landmark SGM.The SGM model contains 8 sloped fault blocks and 19

geologic sequences. Each sequence was split into 1 to 165fault-block-based layers by using proportional, onlap, offlap,and fault truncations. Since SGM layers were arranged faultblock by fault block, a total of 4315 fault-block-based layerswere generated. In order to feed a simulator, the fault-block-based layers have to be converted into model-based layers (orgeologic layers). The converting process is called layerequivalence. Layer equivalence between fault blocks resultedin 1202 geologic layers with 202148 geologic cells per layer(35,934,992 cells for the entire model). There are ten facies inthe geologic model (Table 1). A facies-rule-based uplayeringproperty was developed using Eqs. 4 and 5 with the first-degree difference 2~3, the second-degree difference 0.01, andb=0.8 (see column 2 in Table 1).Figure 8 shows the residual curve of reservoir B uplayering.The first thing to see from the curve is that use of 525simulation layers is good enough to retain all reservoirheterogeneities defined by the facies rules in the 1202-layergeologic model. In other words, reducing 677 layers throughlayer optimization will not change reservoir modelheterogeneity. The residual curve also suggests that use ofabout 50 simulation layers may capture the most importantreservoir heterogeneities of the original geologic model. Theactual simulation model (40 layers) used in simulation historymatch and prediction was constructed by coarsening somelayers without perforations in the upper intervals of the 50-layer model. The 40-layer simulation model resulted inexcellent 15-year history match with little model tuning.

The value of using the proposed method over the traditionalway (uniform lumping) is significant for reservoir Bsimulation. Figure 8 demonstrates that zero residual cannot bereached by using the traditional way unless all 1202 geologiclayers are used. The traditional way also uses more layersthan the proposed method to gain the same residual. Figure 9shows a comparison of the new method and the traditionalmethod on average facies rules per layer by using 24simulation layers. The new method clearly shows a bettermatch of the original geologic model. Even though we usefewer layers for the new method (comparing 500 layers for thetraditional way to 350 layers for the new method), the newmethod still results in a better match of the fine-layer geologicmodel (Fig. 10).Figures 11 and 12 show comparisons of the porosity andpermeability distributions on a cross section between the1202-layer geologic model and the 350-layer simulationmodel created using the new method. Both porosity andpermeability between the two models are similar and nosignificant loss of reservoir heterogeneity is observed from theuplayered model. Figure 13 shows uplayering in differentregions for the reservoir B model. The LGR region (230layers) shows much more details than the coarse grid region(40 layers).Reservoir C is a carbonate reservoir in the Rocky Mountainregion, USA. The geologic model for reservoir C was built by

6 D. LI AND B. BECKNER SPE 62927

using SGM for framework and (RC)2 geostatistic package forproperties. The geologic model consisted of 13 milliongeologic cells with a 240-column by 290-row by 190-layergrid without any faults. There were a total of 16 geologicsequences, each one being split into 1 to 61 geologic layers byusing the onlap, proportional, and truncation options in SGM.Since the facies rules and facies-based relative permeabilitycurves are not available in the course of this study, /hk waschosen as the uplayering property. Figure 14 shows theresidual curves generated by using the new method and thetraditional approach. Improvement on the residual of the newmethod over the traditional way is very significant. Forexample, 125 layers are needed for the traditional way to gainthe same residual of the new method using 25 layers. For thenew method, 25 simulation layers are capable to capture themost important reservoir heterogeneity in the 190-layergeologic model and 58 simulation layers are good enough topreserve most of geologic details. Figure 15 showscomparison of permeability distributions on a cross section forthe 190-layer geologic model and the 58-layer simulationmodel. Very little difference can be observed from the twopermeability distributions. Figure 16 shows the 58-layersimulation model in 3D with permeability displayed in colors.

Conclusions1. The residual curve for uplayering is an important tool for

simulation model construction. From the curve, reservoirengineers can determine the minimum numbers ofsimulation layers needed to capture the most importantreservoir heterogeneity and the entire geologic modelheterogeneity, respectively. Use of the curve also helpsengineers to make their best decisions for simulationmodel construction.

2. The proposed method is very efficient such that allpossible uplayering scenarios for a given very largegeologic model containing more than tens of million cellscan be generated in tens of minutes using the currentordinary computer resources. Engineers are able to easilyextract single or multiple uplayering scenarios for theirsimulation studies.

3. The uplayering property defined in the proposed methodis very flexible. Engineers can use DFC, facies rules, andeven simple properties (e.g. porosity, permeability, orwater saturation) as an uplayering property. Engineers arealso allowed to define their own uplayering property fortheir needs.

4. The proposed method provides engineers not only withautomatic tools but also with some powerful practicaltools. These practical tools include coarsening too thinlayers, splitting too thick layers, using different layers indifferent regions, and tuning simulation layer boundariesto align with nature geologic heterogeneity boundaries.

Nomenclaturea = Scaling factorb = Scaling number

f = Displacing front velocity

k = Permeability, L2n = Number of fine-grid blocks in a specific

directionp = Uplayering propertyp = Mean of uplayering property for a layer

r = Facies rule numberw = Weight (0 w 1)DFC = Displacing front conductivity for a two-phase

flow system, L2FR = Facies rulesN = Number of coarse layersR = Residual between a fine-layer geologic model

and its coarse-layer simulation model fora given uplayering property

S = Saturation = Porosity = Standard deviation of uplayering property for

a layer

Subscriptsi = Index in x direction or initial simulation-layer

schemej = Index in y directionh = Horizontal directionk = Index in z directionv = Vertical directionwi = Irreducible waterx = x directiony = y directionz = z direction

Superscriptsc = Coarse layerf = Fine layer

AcknowledgementsThe authors would like to thank S. Verma, M. Song, D. Stern,and K. Guenther for their review and discussion of this paper.Special thanks go to the Special Projects group, especially K.Wolcott, S. Leininger, and J. Adame. Their discussions andsuggestions of upscaling on the Edop Project lead toimprovement of the proposed uplayering method.

References

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3. Farmer, C.L., Heath, D.E., and Moody, R.O., A GlobalOptimization Approach to Grid Generation, paper SPE21236 presented at the 11th SPE Symposium on ReservoirSimulation, Anaheim, CA., February 17-20, 1991.

SPE 62927 OPTIMAL UPLAYERING FOR SCALE-UP OF MULTIMILLION-CELL GEOLOGIC MODELS 7

4. Garcia, M.H., Journel, A.G., and Aziz, K., AutomaticGrid Generation for Modeling ReservoirHeterogeneities, SPE Reservoir Engineering (May1992), 278-284.

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15. Lake, L. W., Enhanced Oil Recovery, Prentice-Hall, Inc.,Englewood Cliffs, NJ, 1989.

Appendix A Number of ways to group geologic layersAssume that a geologic model contains nz layers and a Nz-layer simulation model will be constructed by grouping thesegeologic layers. There are two rules that have to be followedin layer grouping: (1) one simulation layer must contain atleast one geologic layer, and (2) geologic layers being groupedinto the same simulation layer must follow their original orderand connections. Based on the two assumptions, the totalnumber of ways to group nz geologic layers into Nz simulationlayers can be calculated using the following procedure:1. m= nz - Nz +12 if (Nz 3) { for (j = 1; jnz-Nz; j = j + 1) number[j] = j for (i=Nz-2; i 1; i=i-1) {

for (k=1; k nz-Nz; k=k+1)

{ m = m + number[k]number[k] = number[k] + number[k-1] }}

3. Total number of ways = mFor example, there will be 1.613591026 ways to group 200geologic layers into 20 simulation layers.

Appendix B Displacing front conductivity

For the isothermal flow of oil and water in two immiscible,incompressible phases in a one-dimensional permeablemedium, the displacing (shock) front velocity can be estimatedusing the following equation15:

)( fSD

D Sfdtdx

f

=

(B1)

where

Lx

xD = = Dimensionless distance (B2)

= tD dtALqt 0 = Dimensionless time (B3)L

pAkq h

= = Total flow rate (B4)

L = Total length of the permeable medium in the xdirection

A = Cross section areaSf = Displacing front water saturation

Substituting Eqs. (B2)-(B4) into Eq. (B1) gives:

LpSfk

dtdx

fh

S f

=

)( (B5)

Therefore, we can define the combination of parameters in theparentheses on the left side of Eq. (B5) as displacing frontconductivity (DFC). Table 1 Facies parameters and rules of reservoir B

FaciesID

FaciesRules

DFC(md)

(kh/)1/2(md)1/2

kv/kh Swi(%)

6 0.01 0.0 0.0 0.0 100.0

5 0.02 0.0 2.5 0.0 95.0

3c 3.01 375 11.4 0.001 55.0

3b 3.02 4057 34.7 0.01 20.5

3a 3.03 9092 70.2 0.1 8.9

2 5.01 13890 96.1 0.9 6.5

1b 5.02 15403 91.1 1.0 8.4

1a 5.03 12943 83.5 1.0 11.2

4a 7.01 288 10.0 0.8 45.0

4b 7.02 26 3.0 0.5 80.0

8 D. LI AND B. BECKNER SPE 62927

Figure 1. Uplayering by using a traditional way (uniform lumping)

Figure 2. Uplayering by using the proposed method (residual optimization)

Figure 3. Correlation between permeability and porosity for different facies

0

20

40

60

80

100

0 10 20 30 40 50 60 70 80 90 100

F in e lay e r

C o arse lay e r

Z (f t)

Perm

eabi

lity

(md)

00 1010 2020 3030 4040

0.010.01

0.10.1

11

10

100100

10001000

1000010000

33cc

33bb

33aa

2211bb

1a

4a4a

55

44bbk (md)

(%)

Perm

eabi

lity

(md)

Z (ft)

0

20

40

60

80

100

0 10 20 30 40 50 60 70 80 90 100

Fine layer

Coarse layerResidual

SPE 62927 OPTIMAL UPLAYERING FOR SCALE-UP OF MULTIMILLION-CELL GEOLOGIC MODELS 9

Figure 4. A typical residual curve for uplayering

Figure 5. The residual curve of the reservoir A model uplayering (379 geologic layers)

Figure 6. Uplayering from 379 geologic layers to 29 simulation layers (reservoir A)

Simulation layer

Geologic layer

Number of simulation layers

Res

idual

Number ofgeologic layers

Number ofgeologic sequences

0

A B C

10 D. LI AND B. BECKNER SPE 62927

(a)

Figure 7. Comparison betw(the reservoir A m

Figure 8. Comparison of theway and the new peenod

rero

Tr(b)

the 379-layer geologic model (a) and the 29-layer simulation model (b)el, porosity increases as the color scale goes from blue to red)

sidual curves of the reservoir B model uplayering by using the traditionalposed approach

aditional way

New method

SPE 62927 OPTIMAL UPLAYERING FOR SCALE-UP OF MULTIMILLION-CELL GEOLOGIC MODELS 11

a) Uplayering from 1202 layers to 24 layers (traditional way)

b) Uplayering from 1202 layers to 24 layers (ne

Figure 9. Uplayering comparison of the traditional way and the

Geologic layer

Simulation layer

Simulation layer

Geologic layerw method)

new method (reservoir B)

12 D. LI AND B. BECKNER SPE 62927

(a) Uplayering from 1202 layers to 500 layers (traditional way)

(b) Uplayering from 1202 layers to 350 layers (new method)

Figure 10. Uplayering comparison of the traditional way and the new method (reservoir B)

Geologic layer

Simulation layer

Geologic layer

Simulation layer

SPE 62927 OPTIMAL UPLAYERING FOR SCALE-UP OF MULTIMILLION-CELL GEOLOGIC MODELS 13

(a)

(b)

Figure 11. Comparison of the 1202-layer geologic model cross-section (a) and its offspring 350-layersimulation model cross-section (b) (the reservoir B model, porosity increases as the color scalegoes from blue to red)

14 D. LI AND B. BECKNER SPE 62927

(a)

(b)

Figure 12. Comparison of the 1202-layer geologic model cross-section (a) and its offspring 350-layersimulation model cross-section (b) (the reservoir B model, permeability increases as the colorscale goes from blue to red)

SPE 62927 OPTIMAL UPLAYERING FOR SCALE-UP OF MULTIMILLION-CELL GEOLOGIC MODELS 15

Figure 13. Uplayering in differentlayers in the coarse grid

Figure 14. Comparison of the resiway and the new propo regions (the reservoir B model, 230 layers in the LGR region and 40 region. Porosity increases as the color scale goes from blue to red)

dual curves of the reservoir C model uplayering by using the traditionalsed approach

Traditional way

New method

16 D. LI AND B. BECKNER SPE 62927

(a) (b)

Figure 15. Comparison of the 190-layer geologic model cross-section (a) and its offspring 58-layersimulation model cross-section (b) (the reservoir C model, permeability increases as the colorscale goes from blue to red)

Figure 16. Uplayering from 190 layers to 58 layers (the reservoir C model, permeability increases as thecolor scale goes from blue to red)

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