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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~), [email protected],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
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presented at tie 1986 SPE Armud Techtdcal
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Oct. 5-8:”
3. Haldorscn, H,H. : Wrmdator Parameter As-
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K, K.: “The Effects of PenneabiliV Varia-
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6. Jourml, A. G.: .. Gcowatitics for Conditional
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7. Mantog%au, A. and Wtison, J.: “Simulation
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1987 SPE Amurd Technical Conference and
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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