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Statistical scale-up of reservoir properties:

concepts and applications

Larry W. Lake*, Sanjay Srinivasan

Department of Petroleum and Geosystems Engineering, University of Texas at Austin, 1 University Station Stop CO300, Austin, TX 78712, USA

Abstract

All petrophysical quantities are used at a scale different from the one on which they were measured. This necessitates an

adjustment of the measured values before they are used to develop a reservoir model, a step referred to as scale-up. Scale-up is

complicated by the properties being heterogeneously distributed in space and self- or autocorrelated. The autocorrelation means

that the heterogeneity itself must be scaled up so that the adjusted measurements correctly reflect the property at the coarser

scale.

This paper attempts to understand the uncertainty in assigning scaled up values to finite regions of space. The region may be

a formation interval or a cell thickness for numerical simulation. We use the notion of the variance of the mean of a random

variable to understand the scale-up process. The behavior of the variance of the mean is used to investigate the definition of a

representative elementary volume (REV), and of the behavior of lateral and vertical permeabilities with scale and the resultant

impact on uncertainty distributions for reservoir properties.The paper demonstrates that the notion of the variance of a mean can be used to understand and refine the concept of

representative elementary volume. The change in horizontal and vertical permeability with scale can also be explained using the

variance of the mean using reasonable autocorrelation functions. The impact of scaling up on the autocorrelation structure of the

simulated field is demonstrated in the paper. The results point to the importance of employing rigorous procedures to scale

heterogeneity in order to derive robust estimates of uncertainty.

D 2004 Elsevier B.V. All rights reserved.

Keywords: Scale-up; Variance of the mean; Representative elementary volume; Kriging; Block Kriging; Permeability scaling; Scaling

heterogeneity

1. Introduction

Recognizing and reconciling differences in mea-

surement scales associated with data from different

sources is an important aspect of reservoir modeling.

Typically, the measurement scale (fine scale) is

smaller than the application scale (coarse scale). Since

properties are heterogeneously distributed in space

and self- or autocorrelated, the heterogeneity itself

must be scaled up so that the adjusted measurements

correctly reflect the property at the coarser scale. That

this is so is often overlooked, with scale-up tradition-

ally being done on the properties only, not on their

heterogeneity. As demonstrated below by analytical

calculation and simulation, the intrinsic heterogeneity

plays a role in the scale-up. Furthermore, traditionally

0920-4105/$ - see front matter D 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.petrol.2004.02.003

* Corresponding author. Fax: +1-512-471-3161.

E-mail address: larry _ [email protected] (L.W. Lake).

www.elsevier.com/locate/petrol

Journal of Petroleum Science and Engineering 44 (2004) 27–39

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scaled up quantities and fine-scale measurements can

easily have a different spatial distribution from each

other.

Let Z be a scalar, spatially continuous, Gaussian,random variable distributed in one-dimensional field.

The variance of the average of Z over a distance L is

given by:

Var ð ¯ Z Þ ¼ 2r2

L2

Z n¼ Ln¼0

Z g¼ng¼0

qðgÞdgdn

ð1Þ

where r2 is the variance of the entire field (the

population variance), and q is the spatial autocorrela-

tion function. The variance of the mean should

decrease as L increases; as L approaches zero, the

variance of the mean approaches the population

variance since the averaging volume is now a point.

The variance of Z at a point within L is expected to

increase as L increases (properties within large vol-

umes are more heterogeneous than within small vol-

umes). This is given through Krige’s relationship

(Journel and Hujbregts, 1978):

r2o= D ¼ r2o= L þ r

2 L= D ð2Þ

where ro/ D2 is the variance of Z at a point (o) in a large

volume D, ro/ L2

is the variance of Z at a point in asmall volume L (

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2. Representative elementary volume (REV)

The most fundamental postulate in practical reser-

voir engineering is the existence of the re presentativeelementary volume abbreviated as REV. Fig. 1 illus-

trates the traditional explanation (Bear, 1972).

The plot shows the average of porosity on the y-axis

against the averaging volume V ( x-axis) (any other

petrophysical quantity could be on the y-axis). Begin-

ning at large V , the plot shows that the average of the

property changes as V decreases, becoming stable

(constant) at intermediate V , and then beginning to

fluctuate again as V approaches zero. The REV is the

volume of V at which the small-scale fluctuations begin

when approached from large V . Bear refers to the large-

scale fluctuation as ‘‘inhomogeneity’’ (heterogeneity).

Fig. 1 is the result of a thought experiment because

an actual measurement cannot be done—at least not

over enough scales to confirm the behavior shown in

the plot. V is meant to be a three-dimensional volume.

In this work, L is a one-dimensional proxy for V .

Another way to look at Fig. 1 is in terms of the

variance of the mean of the porosity. Were this to be

plotted on the same scale as in Fig. 1, this variance

would be large at the large scale, zero (or small) at the

intermediate scale and become large again for V less

than the REV. Interestingly, this behavior cannot happen for the autocorrelation models discussed

above, or, for that matter, any two point autocorrela-

tion function that decreases with distance up to a

range and stabilizes to the stationary variance there-

after. The reproduction of the variability at the large

scale is not possible for such a stabilized semivario-

gram. Fig. 2 illustrates.

In Fig. 2, the small-scale variability is 90% of the

total variance, and the large-scale variability contrib-

utes the remaining 10%. The intermediate scale

variability has been omitted. The two remainingscales are clearly visible on the x-axis as the starting

and ending locations of the ‘‘knee’’. The slope of the

plot for L>k3 = 100 approaches 1 as it should for independent Z . As is evident from Fig. 2, the

variance of the mean continually decreases as length

increases and the rate of the decline ultimately

approaches a slope of 1 at length scales greater than the largest range of correlation of Z , in this case

k3 = 100.

We can use the information in Fig. 2 to generate a

plot comparable to Fig. 1 by letting Z be porosity. Z is

now a locally binary (ones and zeros) variable signi-fying pore or grain spaces at points within a volume L.

The average porosity of the volume is simply the

expected value of the binary indicator and the popu-

lation variance of porosity is:

r2 ¼ Var ð/Þ ¼ E ð/Þð1 E ð/ÞÞ ð8Þ

If the random variable Z ¯ is assumed to be Gaussian

with mean of E (/) and variance Z ¯ obtained using Eq.Fig. 1. The representative elementary volume (REV) concept

applied to porosity. Adapted from Bear (1972).

Fig. 2. Variance of the mean of Z for Eq. (5) (normalized by the 2r2

term) as a function of L for a K = 3 model where f 1 = 0.9, f 2 = 0,

f 3 = 0.1, k1 = 0.01, k2 = 1, k3 = 100. The solid straight line has slope

of 1, which indicates spatially independent Z .

L.W. Lake, S. Srinivasan / Journal of Petroleum Science and Engineering 44 (2004) 27–39 29

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(8), then realizations of average porosity Z ¯ can be

generated by Monte Carlo sampling. Fig. 3 shows one

such realization.

Fig. 3 resembles Fig. 2 in that the curve becomesespecially erratic as the averaging distance L

approaches zero. Furthermore, the average porosity

becomes less variable as the volume increases and

approaches the population average for large L. There,

however, is one significant difference between Figs. 1

and 3. There exists no region of stable porosity at

intermediate distance even though the parameters

were chosen (the contribution for the middle scale is

zero and the contribution from the extreme scales are

factor of ten different) to emphasize such stability.

These results are consistent with general observa-

tion. The variability of the average porosity in a set of

1-in.-diameter core plugs is not zero (albeit the

variability is probably less than that shown in Fig.

3), and this variability is certainly smaller than the

variability of porosity on the scale of say 100 Am.

What is surprising, however, is that a substantial

fraction of the observed small-scale variability comes

from large-scale heterogeneity. The large-scale het-

erogeneity, which starts at L =100 in Figs. 2 and 3,

persists to all smaller scales. In other words, the only

way to obtain a stable average is for the averaging

volume to exceed the largest scale of the heterogene-ity. This is an unsatisfactory outcome because it says

that the REV, as defined by Fig. 1, is the largest scale

of heterogeneity in the field. Since it is known that the

largest scale of heterogeneity can be very large—

approaching the field or even basin scale, this defini-

tion suggests that any cell-by-cell computation or anymeasurement is below the REV scale and, hence,

there is added variance (uncertainty) of the modeled

porosity values.

3. Uncertainty assessment accounting for scale-up

The preceding section discussed the systematic

change in the variance of a petrophysical property

with volume. Here, we discuss how scale-up affects

assessment of uncertainty and spatial autocorrelation.

The variance is a measure of the uncertainty in

assigning a property to a reservoir block of certain

volume support. The uncertainty in assigning reser-

voir properties to two blocks is represented through

the block autocovariance between two points a lag

distance h apart. The scale-up of values measured at a

small scale causes systematic changes in the block

autocovariance structure, in a manner similar to the

changes observed above for the variance of the mean

(Frykman and Deutsch, 2002). This means that the

scale-up will affect the assessment of uncertainty in a

property, and this is demonstrated by the simulationsin this section. These simulations yield the uncertainty

of the petrophysical properties caused by the disparity

Fig. 3. One realization of mean porosity for E (/) = 0.3. Same semivariogram parameters as in Fig. 2.

L.W. Lake, S. Srinivasan / Journal of Petroleum Science and Engineering 44 (2004) 27–3930

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in volume support (the value of L above) of the

respective data.

Data integration is often a primary quest of reser-

voir modeling. Several techniques for assimilatingdata from different sources have been proposed (Var-

ela et al., 2002; Datta-Gupta et al., 1995). Data

integration that specifically accounts for the volume

scaling relationships has been less well studied, de-

spite the considerable research effort expended on up

scaling fine-scale geological models to coarse reser-

voir flow simulation grids. Data integration, in the

case of scale-up for flow simulation, consists of using

petrophysical properties that are measured at small

scale (often on a core) support and then analyzing

these data for autocorrelation measures. The measures

inferred from the point support data are used for

simulating spatial variations of the phenomena at

block support. Therefore, the scale-up of both vari-

ance and autocovariance is an issue.

The semivariogram of block support values c̄(V ,V V)

can be expressed in terms of the semivariogram of

point support data c(m,m V):

c̄ðV ; V VÞ ¼ 1

AV NV VA

XV m¼1

XV Vm V¼1

cðm; m VÞ ð9Þ

Eq. (1) expressed the variance of the mean Z ¯ V over

the volume V as a function of the fine-scale semi-

variogram cm(h). Eq. (9) extends that notion of vari-

ance of the mean to the semivariance between the

means Z ¯ V and Z ¯ V V of two blocks; specific compar-

isons between Eqs. (1) and (9) are:

(a) Eq. (9) is written in terms of the semivariance

rather than the autocorrelation function. Eq. (1)

can be converted to this form using Eq. (2).

(b) The origin and destination scales are notateddifferently. In Eq. (1), the origin scale is a point

(o); in Eq. (9), it is a finite volume V V. The

destination scale in Eq. (1) is L; in Eq. (9), it is V .

(c) Eq. (9) is for multi-dimensions whereas Eq. (1)

is in one dimension. The sums in Eq. (9), which

are approximations to integrals, are threefold

integrations.

The discrete sums in Eq. (9) occur because the

continuous reservoir domain is commonly assumed to

be pieced into a discrete set of cells for reservoir

modeling. The cells represent the local or point

support of the property. An important remark pertain-

ing to Eq. (9) is that at a volume support V greater than the REV, the block-to-block autocovariance

c̄ (V ,V V) simply reverts to the prior block variance since

the means of the blocks are then independent of each

other. For any scale less than the REV, the c̄(V ,V V)

value will be finite and different from the variance of

the mean in Eq. (1).

Integration of measured values at a small scale into

the geological model requires that the cross-autoco-

variance (or semivariance) between the small-scale

values and the cell values also be known. These can

be expressed as:

c̄ðV V; mÞ ¼ 1

AV VA

XV Vm V¼1

cðm; m VÞ ð10Þ

Reiterating, if the volume support V is greater than the

REV, the cross-autocovariance between a block and

any point-measured value (left-hand side of Eq. (10))

falling outside the block will identify the prior block

variance.

These scaled up variance and auto/cross-variances

(or semivariograms) can be used to model the spatial

variability of a petrophysical property. The sequentialapproach to stochastic simulation consists of con-

structing conditional cumulative distributions at each

simulation location by Kriging. The mean of the

conditional distribution (assumed Gaussian) is given

by:

Z * ¼Xa

ka ¯ Z a þXb

tb Z b ð11Þ

Z ¯ a, a = 1,. . .,n are the conditioning block data and Z b,

b= 1,. . .,n Vare the measured small-scale data. If only

small-scale data values are available for modeling the

reservoir, the number of conditioning block data n will

be zero. The n + n Vweights ka and tb are obtained by

solving a co-Kriging system:

Xna V¼1

ka Vc̄ðV a; V a VÞ þXn Vb¼1

tbc̄ðV a V; mbÞ ¼ c̄ðV a; V oÞ

Xna V¼1

ka Vc̄ðV a; mbÞ þXn Vb V¼1

tbcðmb V; mbÞ ¼ c̄ðmb; V oÞ

ð12Þ


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The simulation node is represented by V o. The

corresponding Kriging variance is

r2 K ¼Xa

kac̄ðV o; V aÞ Xb

tbc̄ðV o; mbÞ ð13Þ

Assuming the local conditional distribution at V o is

Gaussian with mean and variance given by Eqs. (11)

and (13), a simulated value is obtained by randomlysampling from the Gaussian distribution with mean

from Eq. (11) and variance from Eq. (13). This

simulated value is added to the conditioning data set

for the next node. The simulation is completed by

visiting all the nodes in the simulation domain follow-

ing a random path (Deutsch and Journel, 1998). The

number of conditioning block data, even if zero at the

start of the simulation process will increase as the

simulation proceeds.

The semivariogram model and the conditioning

data used in the example are in Fig. 4. The autocor-

relation is represented by an anisotropic semivario-

gram that has its largest correlation length at 45j to

the x-axis.

Taking the semivariogram model (left in Fig. 4) andthe conditioning data to be representative of point scale

variability as shown on the right in Fig. 4, the objective

is to simulate the spatial variations in porosity on a

scaled up grid. The scaled up grid consists of 10 10cells each of dimension 50 50 ft. Fig. 5 shows onerealization of the scaled up model obtained following

Fig. 4. Semivariogram model (left) and location of point support data in the simulation grid (right) used for demonstrating stochastic simulationusing the scale-up relations.

Fig. 5. The figure on the left is a realization of the simulated field using the volume scaling relations. A fine-scale realization (center) was also

linearly upscaled to give the realization shown on the right.


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the procedure outlined above. For comparison, a suite

of fine-scale models, at a resolution of 50 50 grid blocks, each of dimension 10 10 ft, were simulated

and subsequently upscaled to the 10 10 grid using alinear (arithmetic) aver age of the property Z within

each coarse grid. Fig. 5 also shows one realization of

the scaled up model obtained in this fashion. The fine-

scale simulation assumes that the conditioning data in

Fig. 4 is in fact at the volume support of the fine-scale

grid and hence no scale-up was performed.

Comparing the scaled up simulation results (left in

Fig. 5) with the upscaled model results (right in Fig. 5),

we see that the upscaled model retains the variability

observed in the fine-scale model. This is evident from

the strongly contrasting shades in the pixel map. The

semivariogram corresponding to the upscaled map

(Fig. 5, right) probably exhibits a smaller range of

autocorrelation than that corresponding to the scaled

up simulation map (Fig. 5, left). The autocorrelation

structure (i.e., the heterogeneity itself) is scaled up in

order to generate the scaled up r ealization. In contrast,

the upscaled realization (Fig. 5, right) is obtained by

simply averaging the fine-scale values on a block

support, i.e., the heterogeneity is not scaled up in the

up-scaling procedure.

To demonstrate the effect of using the correct

scale-up relations in the simulation, the average of all blocks in the northeast quadrant of the pixel map

was computed for both the scaled up and upscaled

models. If the quantity Z being simulated is porosity,

this quadrant average serves as a surrogate for the

pore volume in a particular region of the reservoir.

Fig. 6 plots the variability in quadrant average ob-served over 50 realizations of the two models.

The uncertainty of the scale-up realizations is the

smallest. This is expected since direct simulation on

block support uses the scaled up semivariogram with

a longer autocorrelation range (corresponding to

linear averaging), which implies reduction in vari-

ance of simulated values. In contrast, any single

realization obtained by up scaling a fine-scale model

exhibits more variability, i.e., the semivariogram has

a shorter range that causes a wider uncertainty

distribution. However, ergodicity requires the spatial

average of the semivariogram (as computed in the

scale-up simulations) to be equivalent to the average

semivariogram characteristics computed on a large

number of upscaled realizations. This implies that

the two uncertainty distributions depicted in Fig. 5

would converge to the same mean given enough

realizations. However, depending on the range of the

starting point-support semivariogram, the number of

upscaled realizations required for convergence could

be large. This is an important result since it implies

that robust estimates of uncertainty using the fewest

representations of the reservoir model require imple-menting a proper scale-up procedure in stochastic

simulations.

Fig. 6. The variation in volume averages over a portion of the reservoir obtained using a suite of scale-up simulations is plotted on the left. The

variation computed over a suite of upscaled realizations is on the right.


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The simulation approach outlined above is similar

to t he ‘‘missing scale’’ simulation approach proposed

by Tran (1995). In that approach, the rank-order of the point support data are used to simulate the spatial

variability at the block support. That simulation ap-

proach also uses scaled up semivariogram for the

spatial simulation of block support values.

4. A well log example

Another interesting application of the scale-up

procedure using the notion of the variance of the

mean is for analyzing the vertical variability typicallyobserved in well logs. Well logs record changes in

petrophysical properties in the vicinity of a well.

Response variables such as acoustic travel time ex-

hibit different characteristics adjacent to producing

intervals than in non-producing intervals. Fig. 7

shows a typical travel time log recorded over a

1800-ft interval. The log is characterized by relatively

wide depth intervals of high or low acoustic travel

time and also isolated peaks and lows. In a typicalreservoir characterization exercise, logs from adjacent

wells are analyzed and a reservoir model is con-

structed by correlating the log traces at the wells.

Several methods for correlation analysis have been

proposed (Jennings, 1999). Frequently in these tech-

niques, the window size is varied until the maximum

correlation between the traces at adjacent wells

occurs.

The fluctuations observed in the well logs repre-

sent point scale vertical variations in petrophysical

properties. These point scale variations must be scaledup to a REV scale. At the REV scale, adjacent vertical

blocks have attribute values that are independent of

each other. The conjecture here is that autocorrelation

analysis between adjacent wells at this REV scale

would yield a much better idea of the compartmen-

talization of the reservoir.

Fig. 7. A typical acoustic log indicating porosity variations adjacent to a vertical well. The measurement interval is 0.5 ft.

Fig. 8. Vertical semivariogram and model fit computed using the well log data in Fig. 7. The semivariogram fit is shown on a log– log scale on

the right.


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The vertical semivariogram computed using theacoustic log information and the model fit are shown

in Fig. 8. A two-structure exponential semivariogram

model with f 1 = 0.6 and k1 = 6 ft, f 2 = 0.4 and k2 =20 ft

(Eq. 4) was selected.

Using these semivariogram model parameters, a

scale-up plot similar to Fig. 2 can be plotted. The

systematic change in the variance of the mean com-

puted using vertical blocks of increasingly larger size

is shown in Fig. 9 (left). As discussed earlier, at small

scale ( L), the variance of the block averages is large

and about equal to the population variance ro/ D2

. Thevariance decreases as the vertical block size is in-

creased and ultimately the data take on a slope of 1indicating that blocks are at or above the REV scale.

We plot the slope of the variance of the mean plot on

the right of Fig. 9 to better illustrate the approach to

1. Based on Fig. 9, the REV for the well in Fig. 7 isabout 80 ft. The procedure thus yields the optimum

size of the averaging window (80 ft) based on the

vertical variogram model. The averaging window sizewill always be greater than the largest autocorrelation

length but how much more will depend on the point

scale semivariogram structure. If exponential, the

scale will not be much more than the small scale

support. If the point scale structure is linear (non-

stat ionary), the scale will be larger.

Fig. 10 compares the scaled up log against the

original log response. As expected, the high frequency

variations are smoothed by the linear averaging. These

scaled up blocks now become the basis for the

subsequent log-correlation analysis between wells.Well log data was available for seven other wells in

the reservoir. Semivariogram analysis revealed that

wells 2, 3 and 5 exhibit similar point-support semi-

variogram characteristics. Wells 6 and 7 were alike,

but different from wells 2, 3 and 5. The semivario-

gram model parameters are summarized in Table 1.

The volume variance relationships for these wells

are in Fig. 11. As expected, the REV (given by the

Fig. 9. Variance of the block mean (left) and its derivative (right) plotted as a function of block size. The variation in slope of the variance plot is

plotted as a function of the block size on the right. The REV is defined as the onset of the portion with slope = 1.

Fig. 10. Acoustic well log (continuous line) scaled up using a REV block size of 80 ft.


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block size after which the slopes of the straight lines

in Fig. 11 attain a slope of 1) is larger for all other wells since the semivariogram range in general is

larger for the other wells. The REVs range from 80

ft for well 1 to 110 ft for well 4.

The scaled up well logs can now be analyzed for

horizontal correlation. For illustration, the seven

wells were assumed to lie along a cross sectionthrough the reservoir at irregular spacing. The hori-

zontal variograms were computed assuming large lag

tolerances in order to have enough number of pairs

informing each lag. Fig. 12 shows the horizontal

semivariogram calculated on the basis of the scaled

up logs. The sub-REV variations are represented by

the nugget effect. The Gaussian shape (parabolic) of

the semivariogram at short lags indicates strong

lateral autocorrelation (continuity). The horizontal

range is approximately 1250 ft. The horizontal semi-

variogram computed using the original log data (at 0.5 ft measurement interval) has a much larger

nugget effect and a shorter range. The high nugget

effect is directly a result of the spacing of the wells

being large compared to the range of the fine-scale

semivariogram model. Reservoir modeling directly

using the well log information is therefore likely to

yield wider uncertainty distributions.

It is common practice to aggregate well log infor-

mation into property variations within layers. The procedure described above is a viable scheme to

accomplish the task of defining layers in the reservoir

accounting for the pattern of variability exhibited by

the logs. Also note that the issue of how much small

scale variability should be retained in the scaled up

models is important. Perhaps one approach would be

to establish the scaling laws (linear averaging or non-

linear/ power averaging) rigorously through calibra-

tion before performing the scale-up operations de-

scribed above.

5. Average permeability

The variance of a mean can also be used to say

something about the tendency of averages with scale

for properties that have non-Gaussian distributions

such as permeability. This involves investigating the

increase in variance of average permeability with

Table 1

A summary of the semivariogram parameters for all the wells

f 1 k1, ft f 2 k2, ft

Wells 2,3 and 5 0.6 6 0.4 30Well 4 0.6 15 0.4 55

Wells 6 and 7 0.8 20 0.2 40

In all cases, a two-scale autocorrelation model was used.

Fig. 11. Plot showing REV for all the wells calculated on t he basis

of well log data and the semivariogram models in Table 1.

Fig. 12. Standardized horizontal semivariograms computed on the

basis of scaled up (REV) data. For comparison, the semivariogram

computed using the original point scale measurements is also shown.

Fig. 13. Effect of scale on lateral permeability and its distribution.

Adapted from Kiraly (1975).


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scale in the presence of inherent non-linearity in the

scale-up relationships.

Fig. 13 shows how lateral (horizontal or bed

parallel) permeability increases with scale. This ob-servation is among the most common observations in

history matching performance data: well performance

based on core or log-derived permeability must be

increased to match field data. This behavior is at odds

with current upscaling practice wherein the mean

permeability either decreases or remains constant with

increasing scale. The effect is also one of the most

expected types of behavior because there are forma-

tions from which fluids are readily produced, but

corresponding cores from which will pass very little

fluid.

The same scale effect occurs with vertical (bed

normal) permeability except now the vertical perme-

ability decreases with scale as illustrated in Fig. 14.

The vertical permeability decreases by two factor s of

ten as the scale increases by 10. Figs. 13 and 14 are

not directly comparable. Fig. 14 shows the ratio of

vertical to horizontal permeability, wherein, an in-

crease as in Fig. 13, is present in the denominator and

may partially account for the decrease in the ratio. The

x-axis in Fig. 14 is volume, not length as in Fig. 13.

The decrease, nevertheless, is undeniable and is con-

sistent with the common need to reduce vertical permeability from the core to cell scale to effect

history matches in numerical flow simulation.

Both the effects in Figs. 13 and 14 are usually

explained using geology, the vertical permeability

effect, in particular, by the presence of partially

permeable or impermeable shales that are discontinu-

ous. There is, however, a statistical explanation for

both effects as well.

Recall that Z in Eq. (1) is a Gaussian scalar random

variable. Hence, k = e Z

is a log-normally distributedrandom variable. Further, we assume the arithmetic

average (expectation) of k denoted by k H, is a surro-

gate for the horizontal permeability and the harmonic

average of k , k V is a surrogate for the vertical

permeability. Identifying the arithmetic average with

the horizontal permeability and the harmonic average

with the vertical permeability means, of course, that

the calculated properties apply to a uniformly layered

permeability medium. The scale L in the subsequent

paragraphs is perpendicular to these layers.

The non-centered moments of or der j for a log-

normal distribution are given as (Aitchison and

Brown, 1976):

k j V ¼ e j lþ1

2 j 2r2

o= L ð14Þ

In the above equation, l = E ( Z ) = E (ln k ) = l n k G where

k G is the geometric mean of the log-normally distrib-

uted k . ro/ D2 is related to Var( Z ¯ ) from Eq. (2). Hence,

as the scale L increases, the variance of the mean

Var( Z ¯ ) decreases (Eq. (5) and Fig. 2), and the variance

of a point within the scale increases, resulting in

changes in the non-centered moments of a log-normaldistribution.

Eq. (14) for j = 1 yields, the arithmetic average or

horizontal permeability:

k H ¼ k Geðr2 Var ð ¯ Z ÞÞ

2 ð15Þ

The vertical permeability is equivalent to a harmonic

average and since ¯ Z Harmonic ¼ ¯ Z 2Geometric=

¯ Z Arithmetic, Eq.

(14) yields the vertical permeability

k V ¼ k Geðr2Var ð ¯ Z ÞÞ

2 ð16Þ

The vertical to horizontal permeability ratio becomes:

k V

k H¼ eðr

2Var ð ¯ Z ÞÞ ð17Þ

Fig. 15 plots Eq. (15) corresponding to the stable (one

scale) autocorrelation model of Eq. (7). The parame-

ters, shown in the caption, have been adjusted to yield

approximately the same variations as in Fig. 13.

The figure also shows the F 1 standard deviation

of the estimate since this is available from the varianceFig. 14. Variation of vertical to horizontal permeability ratio with

scale. Adapted from Corbett et al. (1996).


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of the mean. The parameters used to generate Fig. 15

are reasonable. The range parameter k =10000 is

greater than the maximum dimension shown with

a= 0.2 being consistent with other literature (Jen-

nings, 1999).The decrease in vertical permeability with scale is

the same as that shown in Fig. 15 but decreasing

rather than increasing; comparing Eqs. (16) and (17).

The ratio of vertical to horizontal permeability is

similarly easy to calculate. Eqs. (15) and (17) provide

a statistically sound approach to scale-up point scale

permeability values before obtaining a history match

in flow simulation. It is quite likely that predictions of

future performance made on the basis of permeability

models that are correctly scaled up will yield robust

estimates of uncertainty.

6. Conclusions

The variance of the mean and how it depends

on averaging scale leads to refinement (if not

abandonment) of the representative elementary vol-

ume concept. It also can explain, at least qualita-

tively, the change in horizontal and vertical

permeability with scale using reasonable autocorre-

lation functions and reasonable parameters for the

autocorrelation functions. This is a consequence of

the various means depending on variability (for

non-Gaussian distributions) and the variability, inturn, depending on scale. Since the averages now

depend on scale, it is possible that the ideas could

be used to generate scale-up factors if the param-

eters of the autocorrelation function can be inferred

from data. The predicted increase in horizontal

permeability is probably greater that what would

be observed in practice as would the predicted

decrease in vertical permeability, mainly because

the ideas were based on uniformly layered reservoirs,

which are idealizations. Nevertheless, the calcula-

tions could form limits for starting points of more

refinement.

It is also apparent that direct scaling up (taking the

averages over volumes) has the possibility of signif-

icantly altering the autocorrelation structure of a

simulated field. In the example shown here, the semi-

variogram of a well log that appears to be dominated

by the nugget effect actually exhibit strong autocor-

relation after scale-up. The well-to-well correlation

became much clearer once the semivariogram was

computed from averaged properties, the averaging

volume being determined from the properties of the

local semivariogram.

Nomenclature

E ( ) Expectation operator

f k Fraction of total variance within scale k

K Number of discrete scales in multiscale

autocorrelation model

k Permeability

k G Geometric mean permeability

k H, k V Horizontal and vertical permeabilities

L Averaging scale

Var( ) Variance operator Z * Kriging estimate of Z

Z ¯ a, Z b Cell mean and point values of conditioning

data at locations ua

and uha Roughness parameter in stable semivario-

gram model

c Semivariance

c̄ (V ,V V) Semivariance between averages over vol-

umes V and V V

v ,v V Point supports within volumes V and V V

k j V Noncentered moment of order j

Fig. 15. Calculated increase of horizontal permeability with scale

calculated with Eq. (15) and the one-scale stable autocorrelation

model. k G = 100, r2 =20, a = 0.2, k =10000.


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ka, tb Kriging weights

kk Autocorrelation length of scale k

el Geometric mean of lognormal distribution

/ Porosityq Autocorrelation function

r2 Population variance

ro/ D2 Variance of a point property within a large

volume

ro/ L2 Variance of a point property with a volume L

r L/ D2 Variance of the average of a property over L

within D

rk 2 Simple Kriging variance

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