5w83=W

New Method for

Reservoir MappingAndre G. Journel, SPE, and Franqok G. Alabert,’ Stanford U

Summary. The sequential indicator

simulation (S1S) algotithm allows

building alternative, equiprobable,

numerical models of reservoir heter-

ogeneities that reflect spatia!-con-

nectivity patterne of extreme values

(e.g., permeability) and honor data

values at their locations. This paper

presents a case study of a sampled

slab of Berea sandstone,

Introduction

A map or, more generslly, a numerical

model of an attribute’s dwhibution in spsce

is rarely sn end goal. Maps are used as input

to some transfer function designed to simu-

late a response of interest (Fig. 1). The

“quslity” of a map m numericaf model

should be appreciated in relation to the p2r-

ticubu transfer function through which it will

be processed. A map is a poor model of real-

ity if it does not reflect those characteris-

tics of the reaf spatial distribution that most

affect the response function.

The goaf of reservoir characterization is

to provide a numerical model of reservoir

attributes (porosity, permeability, satura-

tions, etc.) for input into complex tmnsfer

functions represented by the vtious flow

simulators. The reservoir model is’ ‘good”

if it provides response functions sidar to

those that would be provided by a perfect

model based on an exhaustive sampling of

the rs.sewoir (see Fig. 1). Aspects of the

resemoir that have little influence on the

response of the flow-simulation exercise

considered need not be modele~ however,

reproduction of criticsl reservoir aspects is

essential.

Inmost flow simulations, the single most

influential input is the permeabihw (or trans-

missivity) spatial distribution that conditions

flow paths. 1,2 In a heterogeneous reservoir

involving layers (e.g., shdes/sznds/frac-

tures) with permeability values differing by

several orders of magtimde, flow is condi-

tioned primarily hy connected paths of high

or low permeability values (flow paths and

barriers, respectively). The hktograrn

shape, whether log normal or not, and the

proportions of extreme pemneabilby values

do not matter as much as the spatiaf con-

nectivity of these extreme vslues. Randomly

disconnected small fractures mzy not gener-

ate flow paths, whereas a minute volume

proportion of connected high pei’meabilhy

vslues may control tlow and thus sweep ef-

ficiency and recovery. In such situations,

reservoir charactetiation shotid detect pat-

terns of connected extreme vslues and rep-

resent them in the numericsl model(s) to be

- NOW at El f.Aqultalne,

copyright 1990 Sodely of Petmlmn Enghmm

used for flow simulations. In the following

we srgue that tmdiional mapping critaia,

such as smoothes of the resulting w+’face

or minimum-error varisnce, may not’ be

relevant because they are not related to

reservoir connectivity.

Focusing on spatial connectivity of &z-

treme-valued attributes involves a high

degree of uncertain that must be assessed.

For example, if a given map m numerical

model feature$ a string of high permeabdi~

vslues, is thst string an artifact of the geo-

Iogiczl interpretation or the interpolation 21-

gorirhm used? Assessing spatial uncertainty

is much more demanding than assessing the

local accuracy of aU estimated values along

the string considered. Our solution does not

provide a singIe estimated msp, as in Fig.

lb, but several alternative, yet equiprobable,

maps, as in Fig. lC. All such maps honor

the data values at their locations and repro-

duce a certain number of connectivig func-

tions that model the dependence in space of

the athibute considered map differences im-

age the prevailing spatial uncertainty. Fe+-.

tures fiat appear on SI1 maps are deemed

reliable, and those that appear only on some

maps umelisble, slhough their possible oc-

cumence elsewhere cannot be ruled out. The

reservoir engineer then can make a mmzge

ment decision with some awareness of the

risk imparted by spatial uncertainty.

Finzlly, “hard” data on extreme values

are scsrce or nonexistent and must be sup-

plemented by “SOW’ information. For ex-

ample, because plugs are not taken in shaly

or fractured puts of the core, core plugs fzil

to sample extreme values, thereby biasing

the permeability distriiudon. Log interpreb-

tion, however, can indicate the presence of

snch extreme vslues. Information comes

tlom various sources at various scales with

vstious de~ees of reliabiity ,3’ yet all infor-

mation sources must be accounted for when

dealing with the spatial distribution of ex-

treme vzlues.

Indicator forrmtism allows numericat or

interpretive information to be commonly

coded into elementary bits (valued at zero

or one). These bits are then ,pmcessed in-

dependemly of their origins to generate the

required numericsl models of the reservoti.

212 February 1990. JIW

‘~.

Uwu ,“. –E+*’”’”’””’. .\ -..

---

bn.+), .?. A –y-’’”’”’””.-

IL”.t .““,””’.

-~-”,b,.i.)l.l,.l

!.,,., , *m.

I.Bk.br

Fig. l—Processing reservoir model(s)

into a response distribution.

Berea Data Set

In 1985, Giordano er al.’4 presented a

remarkable work performed on two slabs of

Berea sandstone densely sampled for per-

meability. The corresponding data sets were

given to Stanford U. for additional rezearch,

This study deals with a data set of 1,600 air

perrmameter measurements kken from a

2 x2-ft [61 x61wn] VetiCd slab. These

data, tzlen over a regular, 40x40 grid, rep-

resent an exhaustive sampling of the slab.

Fig. 2 gives the corresponding statistics and

gray-scale map. Note the low variation

coefficient (0,28) and the strong diagonal

banding of the low permeability values,

~ Pra@ice,’ a much smaller sample dataset would be available to reconstitute the im-

age of Fig. 2. A sample of 10 data, at lma-

tions given in Fig. 3, were retained for our

reconstitution exercise. No geostatistical

study is possible with only 10 data point?,;

however, we may borrow permeability vari-

ogr~s from a larger data set taken at the

same scale from a similar depositional en-

vironment. Akema$ively, vzriogram mcdels

may be synthesized from geological” draw-

ings and interpretations. The variograms ca-

lculated from 201,600 data in the directions

along and across banding are shown in Fig.

4. These variogramz are used for the recon-

stitution exemize. (Fundamentals of geosta-

tistical theory and practice can be found in

Ref. 5.)

W 5 shows two recmmbutions. l%. fmt

recomtitution by ordinary !aiSi”g (Fig. 5a)

used the 10 mm.ple data and variogrzms

modeled from Fig. 4, Although the banding

imparted fmm the strong variogam tiso-

~PY was somewhat reproduced, the kriged

map shows the typical spatial smoothing of

all weighted-moving-average interpolation

algorithms. We used an inverse-squared-

elliptical distance-weighting algorithm for

the second reconstitution (Fig, 5b), which

is no better or worse than the first. Here,

the irtverse-squzred-elliptical distance is de-

fined as <h:+ (1 Ohy): and accounts for

the 10:1 anisotropy rmo modeled in Fig. 4.

JPT. February i990

■ 74-,,1.5 md

~ s.,, m,

❑ m.s. s md

❑ ls,-4u.5 md

number of data 1600

mean 55.53

variance 249,

coefficient of variation 0.2L74

minimum 19.5

maximum 111.5

prob. [z s 40.5] 0.17

prob. [40.5 < z .S 55.] 0.36

prob. [55. < Z < 74.] 0.34

prob. [Z > 74] 0.13

Fig. Z-Statistics and gray+cale map of Berea data set.

Fig. 6 gives tie statistics and histogram

of the 1,600 kziged values to be compared

with tie statistics of the 1,600 tme values

given in Fig. 2. The spatial smoothing im-

pazted by kriging is evidenced by tie smaller

variance and deficiency of extreme vzlues:

“17% of the true valuw are <40.5 md and

13% are >74 md, whereas for the faiged

reconstitution they are 4 and 5.%, respec-

tively. ff reproduction of extreme values and

their pattern of spatial connectivity is im-

portant, there is clearly a need for im-

provement.

Spatial Connectivity Measures

lle distribution over the Berea slab of the

N= 1,600 permeability measurements is

denoted by Z(iit), t= 1.. .N, where iit is

the coordinate vector of the MI sample.

Deilne the indicator transform of any vari-

abIe z(u) by

[

1, ifz(~)=z

i(zz)= . . . . . ...(1)

O, if not

The cumulative histogram, FA (z), of the

1,600 data values (i.e., the proportion of

data valued lezs than or equal to a tfue.shold

z) is the arithmetic mean of nonexceedence

indicators of that threshoId.

FA(z) =propofion witbii A of Z( me) <z

where A =tbe set of 1,600 sample locations.

SimOarly, consider theN(h ) pzirs of da~

locatigs separated by the same vector h.

The h-bivariate distcibition is the prop~-

tion of such pairs as Z( =t) and Z( =! + h )

Simldtamsously valued Sz.

FA(~;z) proportion within A o~the pairs

z(=e) <and z(~g+h)=z

= [l/N(~)IEt,{~cF)}i( 7&;z)

Xi(iq+r;z) . . . . . . . . ...(3)

Soft and hard data locations

., . . . . . . . . . . . . . . . . . . . . . . -+..-+.....:00.0 0000.

:’3 O+.OOOOO:

:00. + 0000 :

?~o+ooooo 0.:

~

g ;0000 00+00 :. .

;000+ 0000 0:

:+:0000 0000 :

;. 000 o 0+0 .;, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

, 4,

Fig. 3-Sample data [ooation map + = iO

hard data and o= 63 soft data (constraint

interval). Map unit =0.6 in.

Exhaustive directional Variogram.,,0

,%,

n

● ● . ‘“-w●

2’ ,~~ . ●

Y . .

E ,% ● .s-,.

E*: ,m +++ ++++

++,0++

~, J& h “ ‘o

Fig. 4—Permeability variog rams con-

structed with all 1,600 actual data value=

+ =direction of banding (maximum con-

tinuity), and ● = direction across banding

(minimum continuity). Variogrzm.ordinate

values= md.

with N(z) =the n_mber of location pairs

SUCh that iZ =+ h eA.

213

a

❑ 74-111.5 md

❑ 55-74 md

❑ 45.5.55 md

❑ IWS-C..S md

b

1’

ff~fle goalof reservoir

characterization is to

provide a numerical

model of reservoir

attributes. . . for input

into complex transfer

functions represented

by the various flow

simulators.lY

,,*5

0.,

,.05Ldll“o 20 40

K

—

m ,0,

)

Fig. 6-Statlstlcs and histogram of krlged values.

Ssilnumber of data 1600

mean 55.2

variance 123.

meflicient of variation 0.2

minimum 22.

maximum w

prob. [z s 40.5 0.04

prob. [40.5 < z s 55.] 0.45

wob. [55. < z S 74.1 0.46

prob. [z > 74] 0.05

I

For a given sepxation vector ~and a low

z, the bivgiate ctumdativedistribution fonc-

tion, FA ( h ;z), appears 2.s a measure of con-

nectivity b~ween two low values. Tbe

greater FA(h ;z), the high% the probabilJy

to find a low value Z( ii+ h), a vector ha-

part from .m~nitial low value Z( ii). The

measure FJ (h ;z) is averaged over A and

mzy chmge from one z to another.

A measure of connectiv~il between high

values ofz(ii) andz(ii+h) can bedefmed

by considering a high z and the new indicator

trzn’sform:

. . . . . . . . .. . . . . . . ...(4)

Then, the connectivity measme for high

values is

D.J~;@ =propotion witbinA of_te pairs

2(=)>2 and z(=+fi)>Z

=[l/N(ii)]EtG(~@)l j( 17f;z)

Xj(iir+k’;z). . . . . . ...(5)

Simif3rly, an n-variatz cmmdative-distri-

bution fonciion is the proportion over A of

a given cmllymdon of n dztz jointly vahwd

s z. The inference of such proportion with

n> 2, however, wmdd not be pmctical be-

cause of the lack of enough actually sampled

n’s with a given geometric configuration.

With the 1,600 actual permeability vafues

of the Be~ea data set, th~conneztivity meas-

ures FA(h ;Zp) and DA(h ;Zg) were cafcolat-

ed and plotted in Fig. 7. The fxst Oueshold

is a low vzlue (z =40.5), leaving only

P= 17% of the 1,680 data values below it.

The second threshold is a high value

(zg=74), leaving q=87% of the 1,600 data

vafues below it (i.e., 13% above it)._ Fig.

7a gives the isopletb values of FA(h ;ZP).

The vzlue at the center of the map,

FA (o;zp) =FA(zP) =p =0. 17, is the propor-

tion of permeabdity samples valued below

zP=40.5 m:. The value FA\hl ,h2;zP),

plotted h ~ p~els left and h2 pmels above

the center of Fig. 7a, is the proportion of

pairs_ of permeability samples separated

by h=(hl ,h2) and valued jointly belo~ ZP.

For any given Zp, the PrOpordOn FA (h ;Zp)

y$ties witl tie coordinates (hl ,h2) of

h; that variability is contoured in Fig. 7a.

Simihly, tie va.riab~iv vs. ~=(hl ,hz) of

the proportion DA (h ;Z4), with Zq fixed at

74 md, is contoured in Fig. 7b. Figs. ‘ia and

7b give a global picture of the decrease in

the prObability_Of twO-po@t connectivity as

the distance I h I increases in any paticulu

direction. The anisotropy caused by the

banding observed in Fig. 2 is reflected on

both connectivity maps, although to a much

grzater extent~or the low-valuz-connmdvity

function FA(Lz0,17). This asYmmetrY of

Iow-vs.-high-vzlue spatial connectiviv is at

impmtzot feature. of the Berea data set and

should be reproduced.

The co~ectivity measures FA(~;40.5)

and DA (h;74) calculated on the 1,6CQ

K!ged values of Fig. 5Z are given in Fig.

S. Note that tie threshold values of 40.5 and

74 md correspond to OK 0.04 and 0.95 quan-

tiles of the kdged-value distribution, respec-

dvely, instead of the 0.17 and 0.87 quantiles

of the actuat 1,&3J data distribution. Besides

the overafl deficiency of extreme values

caused by tie smoothing effect of Miging,

it appezrs that the misotropy of the actuzf

connectivity functions shown in Fig. 7 are

poorly reproduced by kriging in Fig. 8.

The comectivi~ measures (not shown

here) calculated from the 1,600 estimated

values of Fig. 5b do not fare better or worse.

than those of Fig. 8 flat correspond to the

kriging dgoritbm. Poor reproduction of con-

nectivity pzttems of extreme values is not

pmticukw to kriging it is a deficiency shared

by afl weighted-moving-average interpola-

tion zfgorithms including spline-fitting, tri-

~8ubItion, and all autoregressive tech.

ruques

Stochastic [maghtg Alternative

Most traditional automatic interpolation al-

gorithms do not account for tie connectivity

measure of Eq. 3 or 5 and, thus, there is

Iittfe chance that they could reproduce any

connectivity pattern.

The conditional simulation algmhhro~8

is designed to genemte alternative, equiprob-

able images honoring the datz values at their

lccations and reproducing a m@el of the at-

hibuto Z covximce. The Z covariance,

Cz(~)=E[Z( E+ ~)Z(iT)]

-{E[Z(Z)]}{E[Z(Z+ 751},

. . . . . . . . ...(6)

214 February 1990. JPT

ab

Fig. 7-Spat[aI.<onneotivity probability measures for the full Berea data set (a) FA(fizO.,,) = connectivity of low values s40.5

md, and (b) DA (fuz0,87) = connectivity of high values >74 md.

.is a measure of average comelation (limar

dependenc~ between any two values z.( Z)

and <( ii+ h), separated by the same vec-

tor, h.’

A covarianc-e measure, however, does not

distinguish among low, medium, and high

z values. in fact, Refs. 5 and 9 show fhat

the Z covariance is .m average of all con-

nectively ftmctions defined by Eq. 3 for all

values of the pair of threshold values z and

z‘. (possibly-different) that apply to Z( z?)

and Z( Zt + h ). Being unique, the Z covari-

ance cannot model different patterns of con-

nectivity, as evidenced by the two different

maps of Fig. 7.

Ifoxdy one connectivity function ‘from Eq.

3 or 5, corresponding to a singk threshold

value, ZP, is to be reproduced, we can use

a Gaussian random fimction model to

genemte the_altem2tive imzges. The cOvar-

iance CY(h )’ of the Gaussian model is

determined such that, tier appropriate clip

p~g at tie p quanttie, ~e resulting ex-

ceedence,indicator field idemifies the

required comectivity model. 10J 1 unfor-

tunately, because the Gaussian mcdel is fully

deten@ed by a single covariance foncdon,

Cr(h ), there is omy one degree of freedom,

and only one comectivity model from Eq.

3 or 5 cm be reproduced. Attempts to re-

prcduce more tim one connectivity function

by clipping the same Gaussian random field

several times 12 are at best approximate.

One solution consists of trading the repro-

duction of a Z ommiance model for the re-

producdo” of~y number K of ccmmectivily

fUINtiOllS~~(/L;Z~), k=l...K@q.30,5).

The resdting equiprobable images will still

honor the same data at their locations. This

1 t

-+~off,,, ,. Ed’w.s, Dr.,,,..

t

-ID

1 t

-20I

-20 -10m,,, in m,,%.,, Di,c,kn

10 20

a b

Fig. 8—Spatial.connectivity probability measures for the kriged data set (a) F~ (~;zo.,s) = connectivity of 10W ValUeS s 40.5 md,

and (b) DA(~;zO,,,) =connectivlfy of high values >74 md.

JFT ● February 1990 2f 5

,74-,11,5 md

b❑ m.,, rnd

a

H w= ‘d

❑ In.5.4a.s md

c d

Fig. 9-Alternative, equiprobable images conditioned to hard data only.

S1S algoriti is described in Refs. 5 and 6.

Fig. 9 shows four alternative images,

generated from tie S1S algorithm, that honor

the same 10 hard data plotted in Fig. 3. All

such images reprcduce the t&e connectivi~

functions of Eq. 3, FA(h ;z?), for ti=

threshold values, Zk, k= 1,2, and 3, hat

correspmd to the 0.17, 0.50, and 0.87 qum-

tiles of the Fig. 2 hislogram. The threshold

values are 40.5, 55.0, and 74.0 md, respec-

tively. These connectivity functions were

modeled after the experimental values that

reproduced are inferred from sample hard

d~a or given a priori as soft structural in-

formation. SOtl structural information can

stem from a lazge amount of hard data from

a similar depositional tivironment (e.g., an

outcrop) or from geologjcd judgment based

on experience andlor digitized tentative

drawings. With a self-repetitive fractal mcd-

el, connectivity models adopted at a scale

where information is available also maybe

extended to larger or smaller scales where

information is sparse. z

The S1S algoridm does not address tbe

problem of inference, nor, does it replace

gccd gmlogical judgment and interpretation.

It allows addbional flexibility to reproduce

spatial-connectivity patterns found, or

thought to prevail, across the reservoir. As

such, the S1S algorithm alfows incorporating

better geological interpretation into tie

reservoir model to be used for flow simula-

tions and reservoir management.

Comectivity functions such u FA ( Z;ZJ

can be inferred from traditional variography

applied to indicator data, i(7,zk), hdicatm,

data we valued at zero or one and thus pre-

sent no outlicr wbq their variogam infer-

ence generally is easier tin fiat Of tie.

original Z attribute, at least for threshold

values z~ that are not too extreme.

were calculated from the 1,600 actuaf dati

values. Fig. 10 shows the repro@ction of

the ~onnectivity functions FA (h ;ZI) and

DA(h ;Z3) as calculated from tie simulated

values in Fig. 9a. The connectivity maps of

Fig. 10 compare well with their counterparts

and mcdels in Fig. 7. The connccdvi@ maps

of Fig. 8, corresponding to the kriged map

of Fig. 5a, area much poorer reproduction.

Honoring Soft Structura~

Informationg

In practice, the connectivity models to be

Honoring Soft Local

Information 13

In most resemoir-ciwacterization exercises,

bird &@ from cored wells or well logs are

sparse but may be complemented by seismic

data. Seismic velocities and amplitudes do

not give direct measurements of such reser-

voir attribwes as porosity or permeability,

but they cm provide additional qualitative

information on,fie atihute value z(ii). 5,14

At ii. on a seismic trace, interval con-

straints on the attribute value maybe avail-

20

‘T

i t

a b

Fig. 10—Spafial~onnectivity probability measures for simulated lma9e ~9. 9* (a) F. (~z,.1,) = cOnne~vitY Of 10w ‘alues ‘m.5

md, and (b) DA(fvzo.m) = connectivity of ~gh values ~74 md.

February 1990. JPT

able z(za)e(a.,b. ). fntemf (aa,bJ may

be obtained by ctilbrating seismic data to

neighboring well data. 1s A shale-fraction

qualitative indicator at Zti also could be de-

rived from seismic data16 and used as soft

porosity or permeabdity information.

fn addition to the 10 hard dam values, sot?

local information under the format of 63

constraint intervals for the pezmezbilty

values were given to improve the reconstit-

ution of the Berea sznd image in Fig. 2. The

locations of hard and sotl information are

shown in Fig. 3, Instead of a hard sample

value Z( Cm ), the soft local infOrmatiOn cOa-

sists of an indicator datum that tels whether

the value Z( 7.) is below the low threshold

VdUe (:.,,7 =40.5 red), above the high

threshold value (Z0.87 =74 md), Or in

between.

Fig. 11 shows four aftenrative images

generated by the S1S afgorithm. These im-

ages honor the same 10 hard data vafues at

their locations (Fig. 3); they afso honot the

63 constraint intervals at theii locations. ‘fbe

locations md connectivity patterns of high

and low permeability values are reproduced

much better-compare Figs. 9 and 11 to

Fig.2.

It is important to use afl available soft in-

formation, whether of structural (e.g., vti-

ograms and anisotropy) or Iocid mture. The

poorer or more fuzzy nature of soft infor-

mation u.wdfy is balanced by its larger sam-

ple size and more systematic field coverage.

Conclusions

A numerical model must show those spatial

features thstt are the most consequential for

flow simulation even if this means ignoring

less important features. Thus, numerical

models may vary for different intended uses.

A reservoir model designed for estimat-

ing net-to-gross ratio and evaluating initial

in-situ satiations need not look like a model

designed for enhanced recovery. However,

the latter model should reproduce spatiaf-

connectivity patterns of extreme pemneabil-

ity values because such patterns wifl affect

the result of a flow-simulation exercise the

most.

fmficator fonnafism consists of commoniy

coding information from various sources

(well data, seismic, geological intwpreta-

tion) into elemenuuy bits (O-1) that are based

on the exceedence of given threshold values.

The spatial distribution of these bits is then

characterized by a series of connectivity

functions orindicator variogmnm

The S1S afgorithm allows expanding the

original indicator information into 2ftema-

tive, equipmbable, numerical models of the

reservoir. These models honor the original

indicator data vzluesat their locations and

reproduce tie imposed connectivity func-

tions; thus, they can reproduce spatizf-vari-

ability patterns that may be different for low

(e.g., flow barriers) mdhigh (e.g., flow

paths oropenfmcmres) values.

Possibly most important, !miicator formzl-

ism affows us to account for soft information

of an interpretive (geology) or fuzzy (e.g.,

seismic datz) mture and, thus, complement

JPT 1990

❑ 74-,!,,s md

❑ ,5.74 md

❑ 40.s-55 md

❑ Is.s.a., md

F[g. ii-Alternative, equiprobable images conditioned tohardand so fl infer-

matlon.

the usuafly sparse hard well dim The result-

ing stochastic images of reservoir heteroge-

neities pmvideus. with spatial-uncertainty

images.

Nomenclature

a. = iow~rbo””d

.& = upper bound

Cr(h_) = wvtimce of Gaussimmodcl

Cz(_h) =Zcovariance

DA(h;z) = comectivity mmsure for

high Z WkS

E= expected value

FA(~;z) = connectivity measure for

low z vafues

FA(z) =cumulative histogram of

1,600 data values

h. =direction along bandin ~

Cartesian system

hy =direction across bandin ~

Cartesiau system

~= bivariate-dstribution

separation vector

K = number of threshold

permeability vafues

n = variate-distribution number

N = number of permeability

measurements (1,600)

N(~) = number of data pirs

separated by h

r = response

ii= coordinate vector

El = coordinate vector of ?th

sample

Fe = 10catio* on seismic trace

Z,Z’ = threshold values

Zk = threshold permeability value

Zp = low threshold value

Zq = high threshold vafue

Z = reservoir attribute

‘(The SE aigorithm

does not address the

problem of inference,

nor do@s it replace

good geological

judgment and

interpretation. It allows

additional fkxibility to

reproduce spatial.

connectivity patterns

found . . . across the

reservoir.”

217

Andr6 G. Journel, a geosfatkticz prOfeS

sor, is chairman of the Stanford U. Ap

plied Earth Sciences Dept. and Oirectol

of, the Stanford Center for Reservoil

Forecasting in Stanford, CA. His re

search includes stoch88tic characterim

tion of rewvoir heterogeneities am

merging hard and sofl data from variou!

sources and scales. Franyais Alabert i!

a resewoir engineer In charge of reser

voir characterization and modeling a

Elf.Aquitzine, France. He holds an M:

degree in geostatistics from Stanford U

and ❑S degrees in geological engineer

ing and applied geophysics from Ecoh

Nat]. Sup&ieure G6010gie In Nancy

France.

Subscripts

A = set of 1,600 sample slab

locations

i = ~&ato* of low ~ values,

such aS 2P

j = indicator of high z values,

snch 8S Zq

Acknowledgments

Funding for this study was provided by the

Stanford Center for Reservoir Forecasting.

The Berea sand data set was graciously

provided by Arco Oil & Gas Co.

References

1. Weber, K. J.: “How Heterogeneity Affects

Oi Rewvew, ” Resemoir Ctimcterizatim,

L. Lake and H. Carroll (eds.), Academic

Press, Washington, DC (1986) 487-544.

2. Hewett, T, A,: ‘+Fractal Distributions of

Reservoir Heterogeneity and Their InR.ence

on Fluid Transport,” paper SPE 15386

presented at tie 1986 SPE Armud Techtdcal

Conference and Exhibition, New Orleans,

Oct. 5-8:”

3. Haldorscn, H,H. : Wrmdator Parameter As-

SigIIMent and the Problem of Scale in Rmer.

voir Engineering,” Reservoir Charactenm-

tion, L, !_&? and H. Ca’roll (eds.), Academic

Press, Washington, DC (1986) 293-340.

4. Giordano, R. M., Salter, S. J., and Mohanty,

K, K.: “The Effects of PenneabiliV Varia-

tions on Flow in Porous Media,vz paper SPE

14365 mesented at the 1985 SPE Annual

Techni&l Conference a“d Exhibition, Las

Ve8as, Sept. 22-25.

5. Jonmel, A. G,: ‘, Fun&rnentds of Geostatis.

tics in Five Lessons,xx Shot! Course in Geolc-

gy, AGU, Washington, DC (1989)8, .1-40.

6. Jourml, A. G.: .. Gcowatitics for Conditional

Simulation of Orebodies,” Economic Geol-

ogy (1964) 69, 673–87.

7. Mantog%au, A. and Wtison, J.: “Simulation

of Random Fields Using the Turning Band

Method,” report No. 26, Dept. of Civil En-

gineering, Mass. Inst. of Technology, Cam-

bridge, MA (1981) 199.

8. Luster, G. R.: “Raw Materiafs for Portland

Cement Applications of Conditioned Sim16a-

tions of Coregionaliiations,’, PhD disse~-

tion, Stanford U. (1985) 531.

9. Alabmf, F.: ‘Stochastic fma5D8 of Spatial

Distributions Using Hard and Soft Informa-

tion,,, MS thesis, Stanford U., Stanford, CA

(1987) 197.

10. Jmu’nel, A.G. and fsaaks, E. H.: ‘,Condhiond

fdcator simulation Application to a Sas-

katchewan Uranium Deposit, ” Maih.

Geo@y (1984) 16, No. 7, 685-718.

11, Vanmarke, E.: Random Fickf.s Analysis and

Symh.sis, fvRT Press, Cambridge, MA

(1983) 382.

12. Matbemn, G. et al.: ‘&Conditional Simula-

tion of tie Gmmemy of Fluvi*Deltaic Reser-

voirs,,, paper SPE 16753 presented at the

1987 SPE Amurd Technical Conference and

Exhibition, Dallas, Sept. 27-30.

13. Joumel, A. G.: “Constmined Interpolation

and Sof! Kriging,,, Pro.., Nth Application

of Computers to the Mineral Industry SP-

Wsium, Sot. of Mining Engineers, Litdeton,

CO (1986) 15-30.

14. Doyen, P.: ‘. Porosity Mappin8 From Seismic

Datx A GwsWkdCd Approach, “ Geophys-

ics (1988) 53, No, 10, 1263-75.

15. Thadaid, S., Abbe% F., and Joumel, A. G.:

“h Integrated GemtatisticatlPanem Recog-

nition Te.bnique for the Characterization of

Re.$evoir Spatial Varhtilit y,”paper present-

ed at the 1987 SEG .4nmual Meeting, New

Orleans, Oct. 11-15.

16, Dumay, J. and Foumier, F.: “Chamcteriza-

tion of a Geological Environment by Mul-

tivariatc Statistical Amlysis of Seismic

Parameters, ” Pro.., SEG.Annual Meeting,

Houston (1986).

S1 Metric Conversion Factors

in. Y. 2.54* E+OO = an

md X 9.869233 E-C-4 = pm>

.Cmversim factor IS e.act.

Provenance

Original SPE lMllW1’iflt, Focustig on SPa-

tiaI Connectivity of Extreme-Vafued At-

tributes Stochastic Indicator Models of

Reservoir Heterogeneities, received for

review Oct. 2, 1988. Paper accepted for

publication Nov. 21, 1989. Revised manu-

script receivid Nov. 2, 19S9. Paper (SPE

18324) first presented at the 1988 SPE An-

nual Technical Conference and Exhibition

held in Houston, Oct. 2-5.

m

218 February 1990 ● JPT

.,.

\ 8muj*r*lwll

lC.

1=1, ”., L

•1IF r&),ulnA) 1=1,

1

.... L

m.

l13!bL● b r

Figure 1: Processing the reservoir model(s) into a respome function

Berea 1600 data aet

g 74.111.s

❑ SS-74

&j 40.s-56

(J 1O.S-4M

number of data 1600mean 55.53

L J

variance 249.coefficient of variation 0.284minimum 19.5‘maximum 111.5prob. [z < 40.5] 0.17prob. [40.5 < z ~ 55.] 0.36prob. [55. < z < 74.] 0.34mob. [z >741 0.13

Figure 2: Statistics and greyscale map of the Berea data set

●✎✎☛

SPE 18324

soft ●nd hud dst- 10 C~tiOXIS

,40 n..: . . . . .. .. . . . . . . . . . . . . . ..+....:00000 o+ o 0:

+;00000000 :

!OOO +000 0:*: :+

:0 +0000000 ::;

:0000 0 0+0 0;&,too 0+00000;:+;00000 000:

+:QOooo 0000:

1 -i ”-”-”----” -”---- ”-”””--”---““”l1 40

Figure 3: Sample data location map

+ -10 hard data0-63 soft data (constraint intervals)

Exhaustive directional variograms

‘w~’250 / N3sW

●%.

~ 200 : ● ●

T o ●

g 1s0 ;0 N57E \●

E ;. + *

~ ,00 . ++++++++

50 ++

I I I I5 lb

& h20

_ pe~-hjfity varjograms

(all 1600 actual data values are used)● across banding (direction of rninurnumcontinuity)+ in the direction of banding (maximum continuity)

0rdin8ry kriging

■ 74-111.s

~ SS-74

❑ 40.5.s5

inverse squmci distance ❑ 10.s-40.s

511

Figure 5: Two reconstituted images using the 10 hard data and exhaustive var-iograms

5a - ordinary kriging

Sb - inverse-squared-elliptical distance weighting

.4*

0.2

0.15

0.0s

o

-20

Ordinary kriging estimates% I I 1 I I I I number of data I 16001

t r20

1

M80 100 120

md)

meanvariancecoefficient of variationminimum .—maximumprob. [z s 40.5] -prob. [40.5 < z ~ 55.]prob. [55. < z s 74.]prob. [z > 74]

Fimre 6: Statistics and histogram of krhwd values

Connectivity of low values

-%% ‘ ‘ “AT

<$-%, -A

1 I 1 1 I t t [D -10 0 10 20

Offset m EOSI-West D,rechon

7a

55.2123.0.222.88.0.04.—@,45

0.460.05

Connectivity of high vulues

7b

Figure 7: Spatial connectivity measures for the full Berea data set

7a - ~A(~ 2.17): COnlledhity of low values s’ 40.5 md

7b - ~A(~ 2.87):Connectivity of high values > ?4 md

. ,**

sPE 18324

Connectivity of low valuesm! 1 I t , t 1 i I

-?0 , , ,- ‘JO -10 0 !0 i

(me! 5“ co,! - Wt%t Q.rdto.

8a

Con[!ectivity of high values, ,

t

-20 -!0 o 10 20IX{*I ,0 Lost-w.st Lkmct,.an

&b

Fimme 8: Smtial connectivity measures for the krigeddata set—-?a - I’A(~; 2.,s) : Connectivity of low values <40.5 rmf

?b -~A(~ Z.M):Ccmectivity of highvalues>74 md

Indimtor simulationswith Iuwddsts

■ 74.11 $.5

❑ 5.5.74

❑ 40,s-55

❑ 10.s-40.s

!2E?s2L‘alternative@Probableimw=g-ratedfrom the SIS akorithmcondhioned to hard data

SPE 18324

Connectivity of high values

‘“~t

Connectivity of low values

1 t’

-20 “ -ion’” 10“II W WIcost%W OWcholl

*

10a

I “----i

t-z”~

cast! m CO,(?WG,IDWCICD”

10b

Figure 10: Spatialconnectivitymeazureafor one S1Simage(upper left imageof figure9)loa - ~@ 2.17): co~=tivity ofh du= ~ 40.5md

10b -~A(~ z.~):Connectivityof highvalues>74 mff

Indicators)mulation$with hard atd softdata

❑ 74.111.s

■ 5s.74

❑ 40!5.ss

Q 1O.5-4M

Figure 11: Alternative equiprobab]ei~ga conditionedto hard and soft infor-mat ion