the ‘missing scale’ and direct simulation of block effective properties

Journal

ELSEVIER Journal of Hydrology 183 (1996) 37-56

The ‘missing scale’ and direct simulation of block effective properties

Thomas Tran Stanford Center for Reservoir Forecasting, Green Earth Sciences Building, Stanford University, Stanford,

CA 943052220, USA

Received 23 June 1994; revision accepted 7 September 1994

Abstract

In reservoir characterization, stochastic simulations are typically based on quasi-point core or well data, yet they are used as cell or block properties whose support is much larger than that of the available measurements. This paper presents some numerical examples to illustrate the problems of neglecting this support effect. A methodology for directly simulating the block effective (equivalent) properties is also suggested. The spatial statistics of the block-support values are obtained from a calibration process which can utilize available outcrop data. Con- ditioning to core-support subsurface measurements is done with a rank-preserving quantile transform.

1. Introduction

Stochastic numerical reservoir modeling involves the generation of several realizations of permeability fields at, ideally, the support and resolution of the data available, say 1 cm for core plug data. Current reservoir flow simulators are incapable of handling reservoir models at such resolution. An upscaling procedure is needed to replace the many spatially correlated cell values constituting a reservoir simulation block by a single tensor representing the block effective (equivalent) property in each major flow direction. (In this paper, we use the term ‘effective’ with a petroleum engineering connotation which means ‘equivalent’ in hydrogeology jargon. Given a reservoir grid block comprising fine scale heterogeneities, its ‘effective’ permeability is that single permeability value which yields the same flow through a homogeneous grid block of the same size.)

The permeability fields are generated conditional to data whose volume support is of several orders of magnitude smaller than the model cell size. The dimension of a

0022-1694/96/$15.00 0 1996 - Elsevier Science B.V. All rights reserved SSDI 0022-1694(95)02821-B

38 T. Tran / Journal of Hydrology 183 (1996) 37-56

typical geostatistical simulation cell is 10 m x 100 m x 1 m whereas core data informs a volume of only a few cubic centimeters. Strictly speaking, cell-centered simulated values inform only the core-sized regions around them since the inferred spatial continuity models and the conditioning data are of core-support. In practice, however, each simulated value is assumed representative of the entire cell and within- cell heterogeneities are usually ignored. The term ‘missing scale’ has been coined to refer to this problem. Upscaling for reservoir flow simulation often starts by assuming that the original data (usually at the core scale) are already representative of a much larger geostatistical model cell.

On the other hand, simulating permeability fields at the core scale is neither practical nor sensible. The huge simulation grids involved would require enormous amounts of computer time and disk storage; moreover, the amount of conditioning data is minute compared with the volume (or area) of the region to be simulated.

In this paper, we will adopt the terminology used by Fayers and Hewett (1993): core or mesoscopic scale (1- 10 cm), geostatistical or macroscale (1 - 100 m), reservoir simulation or megascale (100-1000 m).

This paper proposes a methodology for the direct simulation of (macroscale or megascale) block effective properties. The methodology calls for several synthetic core-support realizations honoring the spatial statistics observed on surface (e.g. outcrop) data. Upscaling is performed on these synthetic realizations to obtain the block effective properties and their spatial statistics (histograms, variograms). The upscaled block properties and their spatial statistics are then entered into a simulated annealing program that generates stochastic realizations of (diagonal) tensors of

Upsuling -

l”fer*“n

-

Fig. 1. Direction simulation of block effective properties using calibration transform.

and rank-preserving quantile

T. Tran j Joumai of zymology 183 f1996) 37-56 39

- Calculate K x

- Calculate K Y

Fig. 2. Using Laplace pressure solver with no flow boundary conditions to calculate block effective permeabilities in directions parallel to the block faces.

Fig. 3. Typical set of realizations used for numerical experiments: (a) permeability core-support realization of size 1000 x 1000, (b) permeability core-support values at the blocks’ centers, (c) block effective ~~~bil~ties in the x-direction, (d) block effective ~eabilities in the y-direction.


block effective properties. Conditioning to local core-support subsurface data is achieved via a rank-preserving quantile transform. Fig. 1 illustrates the steps of the proposed methodology.

2. The ‘missing scale’

In this section, numerical experiments are performed to show the consequences of ignoring the ‘missing scale’. Several core-support stochastic realizations of permeability K were generated. Each realization was partitioned into a number of macroscale blocks each of which contains many simulated core-support permeability values. The simulated values corresponding to the centers of the blocks were also retrieved. The traditional upscaling method using a Laplace pressure solver (Warren and Price, 1961) was used to obtain estimates of macroscale block effective permeabilities in each major direction, & and &, (Fig. 2). Note that the no-flow boundary conditions utilized in Fig. 2 may not correspond to the actual boundary conditions prevailing around each particular block. For each original core-support realization, we obtain three additional realizations: one consisting of the blocks’ central core-support values, and two consisting of the block effective properties in the two directions of the block faces (Fig. 3). The former mimics the current practice of simulating the central values which are assumed to represent entire grid blocks; such simulation will be referred to as the ‘central point realization’ in the rest of this paper. The latter contains effective values that have the correct volume support; it will be referred to as, the ‘block-realization’.

Flow

direction

No-flow boundary

I i i i i i i i i i i i i i i i i i 1

No-flow boundary

Fig. 4. Using Laplace pressure solver with no flow boundary conditions to calculate field effective permeabilities in directions parallel to the block faces.

T. Tran 1 Journal of Hydrology 183 (1996) 37-56 41

For the examples that follow, core-support realizations of permeability were generated. Each realization includes 1000 x 1000 core-support values. These realizations are unconditional in that they were not required to honor any prior data values. Each macroscale grid block consists of 50 x 50 core-support values; therefore, the associated realizations of central core-support values and of block effective permeabilities have dimensions 20 x 20. A Laplace pressure solver was used to estimate the effective permeabilities in the x- and y-directions for each realization take2 as a whole (Fig. 4). We will refer to these as the ‘field effective permeabilities’ K, and &,.

In addition, a two-phase numerical flow simulation was performed for each realization: each 20 x 20 grid represents one quarter of a five-spot waterflooding production/injection scheme. The injector is located near the lower left comer and the producer is located near the upper right comer (see Fig. 5). The production constraints are constant bottom hole pressures at both wells. The field is initially saturated with oil and the porosity is constant at 20%. The three response variables retained are the time to achieve 5% water cut, the time to achieve 95% water cut, and the time to recover 50% of the oil-in-place. The commerical flow simulator ECLIPSE was used (Intera ECL Petroleum Technologies, 1991).

2.1. The Gaussian model

One hundred core-support realizations, each of size 1000 x 1000, were simulated using the Sequential Gaussian Simulation algorithm (Deutsch and Joumel, 1992, p. 164) with an exponential variogram model having a long practical range of 667 grid units in the east-west direction and an anisotropy ratio of 7 : 1. The grey-scale maps and the histograms of the first set of realizations are shown in Fig. 6. As expected from a regular sample, the histogram of the central point realization is similar to that

Fig. 5. Well locations for two phase numerical flow simulation.


1 n b. llNWxlO66 ~Dm-su~DDtt mslimtion tiinber ol Data

mean std. dev.

cwf. of var

0. ‘@ Core%ppoi K (m;

c. 20x20 central CDtB-&WDDDfi m.9lization ‘00. SD.

d. 20x20 central corn--suppoti mtMzation

N”- YlE% iti std. dw. 136

coef. of var 1.32

Core-support K (md)

f. 20x20 block effective Kx maikation Number of Data 400

maan 64.6 0120 std. &v. 63.8

6 co& of var 0.66

_h. 20x20 block Mective Ky maliwtion Numtkr uf Data

mean std. dev.

coef. of var

F6 73:s 1.37

Fig. 6. Typical set of realizations generated by sequential Gaussian simulation and the corresponding histograms for the 90” case: (a) and (b) permeability core-support realization of size 1000 x 1090, (c) and (d) permeability core-support values at the blocks’ centers, (e) and (f) block effective permeabilities in the x-direction, (g) and (h) block effective permeabilities in the y-direction.

T. Tran / Journal of Hydrology 183 (1996) 37-56 43

of the ‘exhaustive’ 1000 x 1000 realization. However, the distributions of the block effective values KX and KY are different from those of the core-support K realizations, particularly in the east-west direction across continuity. Fig. 7 shows a comparison of the field effective permeabilities obtained. Compared with the field effective

Field effective Kx - point realizations Nimber of Data 100

mean 66.2 std. dev. 6.66

n coef. of var 0.11

ora b. Field effective Kx - block maliwtions umber of Data

ozm

2 (11% J $ om C

oaa 0 20

Eff&tive I& (tndr

e. Field effective Ky - block realizations

,f. Field eftective Ky ,

mO,c. Field effective Kx ,

Fig. 7. Field effective permeabilities for the 9o”=Gaussian case: (a) (b) and (c) & for point and block realizations and their scattergram, (d) (e) and (f) K,, for point and block realizations and their scattergram.


permeabilities calculated from the block realizations, those obtained for the central point realizations are smaller for the most-continuous east-west direction and larger for the north-south direction. Imagine that each realization constitutes a reservoir- simulation block consisting of 20 x 20 geostat-sized cells, the current practice of ignoring the ‘missing scale’ is inadequate, because the central point realizations lack the spatial continuity exhibited by the block realizations. Fig. 8 shows the variograms for K, XX and KY; each is calculated by averaging the experimental variograms corresponding to the 100 realizations. Compared with the variograms of the core-support realizations, the variograms corresponding to the block effective properties have lower sills and longer ranges. This is the regularization effect described by Journel and Huijbregts (1978, pp. 77-94), except that the averaging process here is unknown and non-linear.

Fig. 9 shows the results of the ECLIPSE simulations on both the central point and the block realizations. It can be seen that the responses corresponding to the block realizations have slightly lower means and variances. In actual practice, these effects will be much more pronounced because the difference in support involved is many orders of magnitude larger than that (1: 2500) used in this example.

The same exercise was repeated except the direction of maximum continuity was changed to 45” north. In such a case, the grid should be locally rotated to align the grid axes with the major anisotropy directions or a full permeability tensor should be used. However, honoring established practice in the oil industry, we have maintained the diagonal tensor approximation. The first set of realizations with the corresponding histograms are shown in Fig. 10. Fig. 11 shows that the central point realizations tend to yield field effective permeabilities that are smaller than those of the block realizations. Fig. 12 shows the ECLIPSE-derived response variables for both the central point and block realizations for the case where the producer and the injector are aligned in the 45” direction which is the direction of maximum continuity. The block realizations yield responses with lower means and smaller variances; this is due to the fact that the connectivity of high values is better reproduced by the block realizations. Fig. 13 shows the ECLIPSE response variables for the case where the producer and injector are aligned in the 135” direction which is normal to the

0 200 NY) 6M ml 1003 O-?--111 aslmcc Dlrlancc

Fig. 8. Variograms for core- and block-supprt pcrmeabilities (90” Gaussian case). Solid line: point-support y&); long-dash line, block-support 7~. (L); short-dash line, block-support +yxy (L).


direction of maximum continuity. The variances due to the block-realizations are dramatically higher than those coming from the central point realizations; this can be explained by the fact that the tortuous flow paths between the two wells are better reproduced on the block realizations.

Ideally, the above flow simulation results should be compared against true

o sw a. 5% water cut - pht realizations o m b. 5% water cut - block reaikafiona Number of Data 100 Number of Data 100

mean 6.82 mean 6.56 oua_ std. dev. 4.11 ouM_ std. dev. 3.50

2‘ coef. of var 0.60 6

coef. of var 0.53 C 03% C Qm- 0 &

J 2 0x0: z 02-m: u. : t :

om_ 0 IW-

o 160 d. 95% water cut - block reaMzatlons Number of Data 100

mean 66.1 mean 60.3 std. dev. 35.0 std. dev. 28.2

coef. of var 0.53 6 cwt. of var 0.47

5 S OoBo S! LL

0250 e. 50% oil produced - point realizations Number of Data 100 Number of Data 100


G coef. of var 0.52 coef. of var 0.47

C 0150 J $ (r*c0 t

0 Time

Fig. 9. Flow responses of point and block realizations for the 90” Gaussian case: (a) and (b) time to reach 5% water cut, (c) and (d) time to reach 95% water cut, (e) and (f) time to produce 50% of the oil.


a. 1 1tXNbf10Wcotesqport1~Mzatlon

Num-tlE :F std.dw. 133

f Q.‘w caef. ol var 1.31

8 = a,m

t

C. wntral corsauppmt mMz8tion Oam NumberotData 400

mean 33.6 std. dew 133

$ o’m coot. of var 1.34

9 earn P L

oaa

u. 2 block dfective Kx malizatlon Number cd Data 400

nman 72.6 std. dw. 30.4

coekofvar 1.11

block dhctiva Ky tdizHon

sm. Number of Data 400 mean 72.6

std.dev. 30.8 coef.ofvar 1.11

en #I m.0 w.0 00

Fig. 10. Typical set of realizations generated by sequential Gaussian simulation and the corresponding histograms for the 45” Gaussian case: (a) and (b) permeability core-support realization of size 1000 x 1000, (c) and (d) permeability core-support values at the blocks’ centers, (e) and (f) block effective permeabihties in the x-direction, (g) and (h) block effective permeabilities in the y-direction.


responses or a set of reference responses, e.g. responses derived from ECLIPSE simulations performed on the realizations with full resolution 1000 x 1000. Our workstations are not yet capable of handling these million-cell models; we plan to run a few of these models on a CRAY to gauge the computer memory and time

(13m a. Field effectiva Kx - point realizations -1 n Number of Data 100

~ m b. Field effectiva Kx - block realizations -f Ih Number of Data 100

Effective Kx (md) Elfective Kx (md)

,,p. Field effective Kx ,

~xa d. Field effective Ky - point malizations 8. Field e ive oaJ Ky - block raelizations

Number of Data 100 1 “ilT Number of Data 100 mean 34.0 mean 37.5

std. clav. 3.64 std. dev. 3.11

Effective Ky (md) Effective Ky (md)

,,,f. Field effective Ky ,

Fig. 11. Field effective perrneabilities for the 45”_Gaussian case: (a) (b) and (c) & for point and block realizations and their scattergram, (d) (e) and (f) KY for point and block realizations and their scattergram.

48 T. Tran / Journal of Hydrology 183 (19%) 37-56

required. It can be argued from theoretical grounds, however, that the block realizations yield responses that are more accurate: the block realizations consist of values that have the correct support, hence the corresponding flow simulation results should be more accurate.

b. 5% water cut - block realizations

.95% water cut -point realizations o,~ d. 95% water cut - block realizations Number of Data 100 Number of Data 100


6 coef. of var 0.36 oz coef. of var 0.33

5 5 3 o.wo $

g 0°80

t c LL

Time Time

ocm 6

50% oil DrOduced - DO/nt reSliZStiOnS Number of Data

mean std. dev.

coef. of var

Time

100 51.7 - 22.6 0.44

-

50% oil produced - block realizations Number of Data 100


coef. of var 0.35

Time

Fig. 12. Flow responses of central point and block realizations for the case where direction of maximum continuity is 45” and the wells are aligned in the same direction: (a) and (b) time to reach 5% water cut, (c) and (d) time to reach 95% water cut, (e) and (f) time to produce 50% of the oil.

T. Tran / Journal of Hydrology 183 (1996) 37-56

3. Direct simulation of block properties

49

This section outlines the proposed methodology for directly simulating the block effective permeabilities. This is a new area of research; the following description of the

water cut- point malizetiona Number of Data 100


coef. of var 0.48

50 %lt? 1Yl 203.

o loo b. 5% water cut - block realIzatIona Number of Data 100

o.am mean 0.34 std. dev. 10.3

6

own coef. of VW 1.23 5 OUYI

g 0x0 E LL 0200

%

Time

o,o d. 95% water cut - block realizations Number ot Data 100 Number of Data 100


coef. of var 0.44 ox coef. ot ver 1 .OO

5 2 Pm CT P IL

!co. IY) 0 Time

0x.l e. 50% oil produced - point realizations om f. 50% oil produced - block reallretlons Number of Data 100 Number of Data 100

mean 45.3 mean 44.4 Olso std. dev. 20.9 std. dev. 46.7

oz coet. of var 0.46 6 coef. of var 1 .Q5

s 3 (IIW 5 ZI 01w B $ Ii Ii

Fig. 13. Flow responses of central point and block realizations for the case where direction of maximum continuity is 45” and the wells are aligned in the normal direction: (a) and (b) time to reach 5% water cut, (c) and (d) time to reach 95% water cut, (e) and (f) time to produce 50% of the oil.


methodology is kept at a fairly general level as details of implementation are still being worked on.

3.1. Methodology

Block effective permeability not only depends on the heterogeneity within the block, but also on the actual boundary conditions existing at the block interfaces. Since the averaging process for permeability upscaling is not known analytically except under very limiting conditions all involving a multivariate Gaussian random function, the spatial distribution of block effective permeabilities cannot be inferred analytically from those of core-support data. This suggests that the statistics of the block-support effective permeabilities be inferred from numerical averaging performed on core-support calibration images having the same statistics (not necessarily Gaussian-related) as those of the available data (e.g. outcrop data, core data). This is the approach originally proposed by Gomez-Hernindez (1991). Any upscaling technique of choice can be used for the numerical averaging process. The quality of the inferred block statistics depends, of course, on the quality of the selected upscaling process and of the representativity of the core-support data used to build the calibration images.

Once the block statistics are obtained, traditional geostatistical techniques can be applied to generate equi-probable realizations of block effective permeability tensors (usually assumed to be diagonal).

The next question is how to obtain block-support conditioning data from the available core-support subsurface data? Since the support of the core data is dwarfed by the size of the macroscopic blocks, it can be argued that the core data should only be used as a ranking index at the block scale, i.e., a block whose core data are high on average should have a high effective property, and vice versa. Based on this premise, the block conditioning data can be obtained via a rank-preserving quantile transform, see hereafter.

3.1.1. Calibration The first step calls for calibration of the block statistics. L core-support

unconditional realizations honoring the statistics of available core data are generated. These realizations are unconditional in the sense that they do not honor the subsurface core data at sampled locations; only the sample statistics (e.g. histogram, variogram) need be reproduced. If high-resolution outcrop data are available, they can be used in the generation of calibration images. The area1 (or volumetric) extent of such calibration images should cover several macroscopic blocks so that sufficient data are available for the inference of block spatial statistics, e.g. the first few lags of their variograms.

A numerical upscaling technique is applied to obtain the effective permeabilities for each block. In this paper, the Laplace single-phase steady-state flow equation, V(KVP) = 0, with no-flow boundary conditions is solved numerically to estimate the block effective permeability in each major direction (Warren and Price, 196 1; Aziz


and Settari, 1979, p. 358). For the 2-D case, we now have L unconditional realizations of K, and K,,.

3.1.2. Direct simulation of block permeability tensors by simulated annealing In this paper, simulated annealing was used to cosimulate I?, and I$ honoring the

auto-variograms r~~,g~((“), TK~,E~((A), and the cross-variogram r~%,a (L) for a certain number of lag values (A(. These variograms would be calculated from the block calibration images obtained in the previous step. Note that, with simulated annealing, the users have the option of not modeling these variograms; the target variograms can be calculated from the calibration images at the beginning of the program. There are three components in the objective function corresponding to the three previous variograms. It is possible to add more components to the objective function (Deutsch, 1992; Deutsch and Cockerham, 1994) to reproduce multiple point statistics and connectivity of extreme values (Journel and Alabert, 1989).

3.1.3. Conditioning to local core-support data As mentioned above, the simulated block effect values should be conditioned to the

5m a. Kx vs arithmetic

1

e (9Odeg)

correlation 0.999 rank correlation 0.999

Keff-X (md)

e (9Odeg)


Keff-X (md) Keff-Y (md)

5.m b. Ky vs harmonic average (9Odeg)

Sm E Fw correlation 0.998 m

: rank correlation 0.998

0 ):

‘g 2m E :-

E z Irn I

5Dl d. Ky vs geometric avera 1 P

e (90deg)


Fig. 14. Scattergrams of block effective values vs. some power averages for the Gaussian model with 90” anisotropy: (a) Kx vs. arithmetic average, (b) & vs. harmonic average, (c) Rx vs. geometric average, (d) KY vs. geometric average.


spatial ranks of the core-support data. The block-support conditioning data can be obtained as follows.

(1) For each block containing enough core-support data, calculate some ‘average’ (e.g. geometric) of all its internal core-support data.

(2) The distribution (histogram) of these averages can be transformed to identify the histogram of the effective permeabilities as obtained from the calibration step. This operation uses a rank-preserving quantile transform (see Deutsch and Joumel, 1992, pp. 209-214).

Once the block-support conditioning data are obtained, conditioning is easily done with simulated annealing: the blocks with conditioning data are simply not perturbed during the annealing process.

Fig. 14 gives the scattergram of the (20 x 20 x 100 = 40 000) block effective values (RX and &,) vs. some power averages of the within-block core-support values, for the Gaussian model with 90” anisotropy. The rank correlations of & with the arithmetic mean and of KY with the harmonic mean are almost one as expected from laminar flow. The rank correlations of both & and KY with the geometric mean remain above 0.98. Fig. 15 gives the scattergram of K, x KY vs. the geometric average of core- support values for the Gaussian model with diagonal anisotropy: the rank correlation exceeds 0.99. It thus appears that conditioning simulation of the block effective values to the ranks of the geometric averages of the within-block data would be as effective as having those blocks’ actual effective values, provided of course that there are enough of such internal data within each of the blocks retained for conditioning.

Instead of the geometric average, one could have considered another power average to accommodate anisotropy of the within-block heterogeneities, for example, an arithmetic average in a direction parallel to lamination or bedding (Deutsch, 1989). It has been our experience, however, that ranks of various power averages do not differ significantly, hence the geometric average best suited for the isotropic case might be retained.

correlation 0.993

0 ‘O” K%X (:d)

Fig. 15. Scattergrams of Rx c KY vs. geometric average for the Gaussian model with 45” anisotropy.

T. Tran 1 Journal of Hydrology 183 (1996) 37-56

~~~on*0126sampkb~k 53

0 0

0

Fig. 16. The 2500 x 2500 reference image and the locations of 25 blocks containing core-support conditioning data.

3.2. Application on synthetic data

A synthetic reference data set was generated with the same variogram model as that used in the previous 100 realizations. The reference image consists of 2500 x 2500 = 6.25 x lo6 core-support values; therefore, this reference image is composed of 2500 macroscale blocks each of which consists of 50 x 50 core-support values. Twenty-five of these 2500 macroscale blocks were considered informed by the ranks of the geometric average of 25 internal core-support data per block (see Fig. 16). Using block statistics deduced from the previously described calibration, 100 realizations of the block effective property (Rx x KY, since the anisotropy is 45”) conditional to the pre_vious 25 rank values were generated. Calculation of field effective permeability K, and flow simulation were applied gn each of these 100 realizations yielding the statistics of Figs. 17 and 18. True K, and true reservoir responses were obtained by performing the same c_lculations on the reference image.

Fig. 17(a) gives the predicted field value K, using the central point value (traditional approach ignoring the ‘missing scale’), while Fig. 17(b) gives the

a. F/e/d effective Kx - Pofnt realkatlons b. Field et7ectlve Kx - Block nral&ations Number of Data 100 Number of Data 100


E coet. of var 0.04 6 coef. of var 0.03

gj ow 5 ocw

& S 2 2 u IL

0c.x omo 300 240

Field effect?ve “Ki (md) x)0 360

Field eff e&e ii (md)

Fig. 17. Distributions of predicted field effective permeability & for the reference image: (a) using the central point values, (b) using directly simulated block effective pexmeabilities. The dots indicate the ‘true’ value.

54 T. Tran / Journal of Hydrology I83 (19%) 37-56

predicted value of & using the proposed approach by direct simulation of block effective property. The reference (true) value (36.5) is indicated by the dot. The probability dist~bution provided by the proposed approach appears not only more accurate (better centered on the actual value) but also more precise (less spread) than that provided by the traditional approach.

a. 5% water cut -point realizations b. 5% water cut - black re8liza~ions

std. deb. 66.7 coef. ot var 0.53 z

: z L I.!.

std. dsv. 39.7 coef. of var 0.37

Time . Time

c. 96% water cut - point reallustlons d. 95% water cut - block raatizations Number ot Data 100

mean 1115 std. dev. 653 std. dev. 261

$ oixl coef. of var 0.60 6 DIZO cuef. of var 0.32

t z g oc%a

za w o.osa

h p! u.

00&l

0 Time . lime

e. 50% oit produced - point real~at~~s f. 50% oil prockmd - block realist Number of Data 100 _-_ I Number of Data 100 0.x.?

mean 666 mean 616 std. dev. 439 II std. dev. 193

coat otvar 0.54 --.T coat. of var 0.31

l Time . Time

Fig. 18. ~s~butions of predicted flow responses for the reference image using ~aditio~l and proposed approaches: (a) and@) time to reach 5% water cut, (c) and (d) time to reach 95% water cut, (e) and(f) time to produce SO% of the oil. The dots indicate ‘true’ responses.


Figs. 18(a), (c) and (e) give the flow simulation results for the traditional approach assimilating block values to their central point values. Figs. 18(b), (d) and (f) give the corresponding results for the proposed approach. Again the latter results appear both more accurate and more precise (note the halving of standard deviations).

4. Conclusions

In this paper, using numerical examples, we have shown the impact on flow responses of ignoring the heterogeneities within the modeling cells or blocks. The difference in supports shown in the examples (2500 : 1) is many orders of magnitude less than one would encounter in actual practice; hence the differences in flow responses between block vs. point realizations would be much more severe.

Next, we have proposed an approach for a direct simulation of block effective properties that accounts for the difference in support volumes of the conditioning core data and the block properties. This approach offers a practical alternative to the cost-prohibitive multiple high-resolution simulation at the core-scale followed by a scale-up process.

The block spatial statistics can be inferred from the scale-up of core-support realizations based on densely sampled outcrop data. Although any other pixel- based geostatistical technique can be used to simulate the block properties once the block statistics are inferred, we suggest using simulated annealing because it can work directly with the block calibration images and it offers the flexibility of further constraining the realizations to honor other types of information (e.g. high-order statistics, soft data).

Although preliminary results based on synthetic data appear promising, more studies are required to provide a complete evaluation of the proposed methodology for upscaling. Future works will include the integration of both hard and soft data of different supports.

References

Aziz, K. and Set&, A., 1979. Petroleum Reservoir Simulation. Elsevier Applied Science, New York. Deutsch, C.V., 1989. Calculating effective absolute permeability in sandstone/shale sequences. SPE

Formation Evaluation, 4: 343-348. Deutsch, C.V., 1992, Annealing techniques applied to reservoir modeling and the integration of geological

and engineering (well test) data. Ph.D. Thesis, Stanford University. Deutsch, C.V. and Cockerham, P.W., 1994. Practical considerations in the application of simulated

annealing to stochastic simulation. Math. Geol., 26: 67-82. Deutsch, C.V. and Journel, A.G., 1992. GSLIB: Geostatistical Software Library and User’s Guide. Oxford

University, New York. Fayers, F.J. and Hewett, T.A., 1993. A review of current trends in petroleum reservoir description and

assessment of the impacts on oil recovery. SCRF Rep. No. 6, Stanford University, Stanford. Gomez-Hemandez, J.J., 1991. A stochastic approach to the simulation of block conductivity fields

conditioned upon data measured at a smaller scale. Ph.D. Thesis, Stanford University.

56 T. Tran / Journal of Hydrology 183 (19%) 37-56

Intera ECL Petroleum Technologies, 1991. ECLIPSE 100 Reference Manual. Intera ECL Petroleum Technologies, Highlands Farm, Greys Road, Henley-on-Thames, Oxon, UK.

Joumel, A.G. and Alabert, F., 1989. Non-Gaussian data expansion in the earth sciences. Terra Nova, 1: 123-134.

Journel, A.G. and Huijbregts, C.J., 1978. Mining Geostatistics. Academic, San Diego, CA. Warren, J.E. and Price, H.S., 1961. Flow in heterogeneous porous media. Sot. Pet. Eng. J., 1: 153-169.

the ‘missing scale’ and direct simulation of block effective properties

Documents