Supported by

Applied Mathematics Subprogram of the U.S.Department of Energy DE-FG02-90ER25084,

Geosciences Program of the U.S.Department of Energy DE-FG02-92ER14261,

Oak Ridge National Laboratory subcontract 19X-SJ067V,U.S. Army Research O�ce through the Mathematical Sciences Institute

of Cornell University under subcontract to SUNY Stony BrookARO contract DAAL 03-91-C-0027,

National Science Foundation grant DMS-9201581.Conselho Nacional de Desenvolvimento Cienti�co e Tecnologico, Brasil

September 1992

AN ANALYSIS of FIELD DATA

PERMEABILITY

Leonardo Fonseca, James Glimm,

and W. Brent Lindquist,

SUNYSB-AMS-92-16

i

An Analysis of Field Data Permeability

Leonardo Fonseca 1

James Glimm 2

W. Brent Lindquist 3

Department of Applied Mathematics and StatisticsUniversity at Stony Brook, Stony Brook, NY 11794{3600

Abstract

Spatially heterogeneous permeability data from several sites is reviewed. The

spatial correlations in this data are valid (i.e. statistically signi�cant) for short

distances only. Multifractal power-normal distributions �t this data about as well

as the more commonly used exponential models, while better for long distance

correlations inferred indirectly from hydrological studies, with the data combined

between distinct sites. The data reviewed, when interpreted in the power-normal,

multifractal model, is consistent with our earlier theoretical prediction that the

fractal scaling exponent � satis�es � 2 [0; 1].

AMS (MOS) Subject Classi�cation: 82A32, 76S05

Key words: porous media, heterogeneity, permeabilitya1 Supported by the Conselho Nacional de Desenvolvimento Cienti�co e Tecnologico,

Brasil2 Supported by the Applied Mathematics Subprogramof the U.S. Department of Energy

DE-FG02-90ER25084, The U.S. Army Research O�ce through the Mathematical SciencesInstitute of Cornell University under subcontract to SUNY Stony Brook, ARO contractnumber DAAL 03-91-C-0027, and the National Science Foundation, grant DMS-9201581

3 Supported by the Applied Mathematics Subprogramof the U.S. Department of EnergyDE-FG02-90ER25084, the Geosciences Program of the U.S. Department of Energy DE-FG02-92ER14261, and Oak Ridge National Laboratory subcontract 19X-SJ067V.

1

1. Introduction. The purpose of this report is to review data concerning spatial

heterogeneity of permeability from certain well characterized geological sites. Based on our

earlier theoretical and computational studies of hydrological dispersion and a comparison

to hydrological dispersivity �eld data, we have preferred a multifractal model of geology,

for the description of heterogeneities at large length scales. The data reviewed here applies

to an entirely disjoint range of short length scales, typically less than one meter. For this

short range geological data, we see that the multifractal models appear to apply about as

well as, but not better than, �xed length scale models such as exponential distributions.

An additional conclusion is that the data is not su�cient to distinguish between

the various models. Because there is considerable interest in scale up, or comparison of

dispersion across multiple length scales, we also comment on methods of data acquisition

and interpretation which would be useful for multiscale studies of geological heterogeneity

and hydrodynamic dispersivity.

2. Multifractal, Power-Normal Distributions Recent work [1,3-6] has investi-

gated a statistical theory for the spatial distribution of permeability in a porous medium.

The model incorporates a continuum of length scale behavior in systematic but not nec-

essarily self-similar manner. The basic assumption, which speci�es the range of validity of

the model, is that measured geological data be slowly varying in logarithmic variables, i.e.

log{log plots of the data with distance are assumed to be smoothly varying, having well

de�ned tangents with nonuniversal slopes. Speci�cally for the permeability, K, we adopt

a power{normal distribution model, where

� =

8<:Kp � 1a

p; p 6= 0;

log(K); p = 0;

(1)

is a normally distributed variable with mean value ��. The power{normal model allows

for a continuous change of the distribution function, from normal (p = 1) to log{normal

(p = 0). We assume �� � � � �� has a two{point correlation function obeying the above

basic assumption. This allows a meaningful parametrization,

h�� ��i(r) = b�r��(r) ; (2)

2

of the two{point correlation structure as a function of spatial separation r. The brackets

h�i denote ensemble averaging. The exponent � is assumed to be a slowly varying function

of log(r). This parametrization assumes a scalar permeability �eld with stationary hetero-

geneity that is either isotropic or is derived by a nonisotropic change of length scale from

an isotropic �eld. Further generality, not considered here, arises from the use of a tensor

permeability.

For single phase ow, the isotropic permeability �eld will produce a velocity �eld whose

deviation, �v � v��v, from mean ow �v along a streamline has a correlation function which

can be parametrized in the same form

h�v �vi(r) = bvr��(r) ; (3)

This model has been applied to single phase ow whose movement is monitored by the

continued injection of uid tagged by a passive tracer of constant concentration into a

medium of initially untagged uid. A perturbation theory applied to this velocity model

[2] has shown that, up to error terms of order �v4, at a �xed travel time, t, the one

dimensional tracer concentration, averaged over an ensemble of ow streamlines, has a

pro�le determined by an advection-di�usion equation. The di�usive segment of the pro�le

is termed a mixing zone and measures the dispersive character of the ensemble of velocity

streamlines. The width l(t) of the mixing zone can be parametrized in the form

l(t) = blt (t) : (4)

The time asymptotic growth rate of the mixing zone is [2]

(t!1) = max

�1a2; 1� �(r !1)a

2

�; (5)

The �nite time behavior of the mixing zone can be theoretically determined [2] in terms

of the velocity �eld correlation alone, as

l(t) = 2

�Z t

0

(t� t0) h�v �vi(�vt0) dt0�1=2

; (6)

(t) =tR t0h�v �vi(�vt0) dt0a

2R t0(t � t0)h�v �vi(�vt0) dt0

: (7)

3

The theory predicts (t) � 1=2. The Fickean limit (t) = 1=2 is shown to occur only if

the velocity �eld is su�ciently uncorrelated. With signi�cant correlation, a non-Fickean,

> 1=2, behavior is predicted.

In [3-6], the predictions of this model were veri�ed by numerical computation even for

heterogeneities of moderately large amplitude. Beginning with an ensemble of numerically

generated, isotropic, scalar permeability �elds, characterized by a distance independent

exponent

�(r) = const ;

the velocity �elds for each realization were obtained by mixed �nite element solutions of

the elliptic system determined from Darcy's law together with the assumption of an in-

compressible single phase uid. Tracer movement for single phase ow was numerically

computed for each permeability/velocity realization by solving a two-phase, �rst contact

unit mobility ratio miscible ow model. The two point velocity correlation function, aver-

aged over the ensemble of streamlines for the velocity �elds, was then determined and the

prediction (7) compared to the the result from the ensemble average of the numerically

computed ow equations. Two values of � were investigated, � = 1 characteristic of a

completely uncorrelated permeability �eld and for which Fickean growth of the mixing

zone is predicted, and � = 0:5, for which short time transient e�ects are signi�cant, and

non-Fickean growth is predicted.

Field results on apparent dispersivities, see, for example, the compendium presented

in Figure 1 of [7], lend support to non{Fickean growth (i.e. � 2 [0; 1]) for typical geologic

media.

To complement the existing numerical work described above, and to motivate future

work, it is necessary to determine typical values or ranges of values for the permeability

exponent �. Although the numerical work described above was performed with constant

�, it is clear that the description of geologic media will be improved with an r{dependent

�. In this report, we examine several assemblages of permeability data available from or

described in the literature, and perform �ts of the form (2) to determine actual character-

izations of �(r). The permeability data sets used are:

- eolian deposit data (Page Sandstone of Jurassic age) [12],

4

- sand and gravel aquifer data from the Borden [8,9] and Cape Cod [10,11] sites,

- fracture dominated geologic data from the Wag6 site on the Oak Ridge Reservation

[15].

In the following sections we deal with each of the data sets.

3. Page Sandstone Data. The Page Sandstone formation consists of cross{bedded,

quartzose sandstone, extending as a linear sandbody from north{central Arizona across

much of southeastern Utah. Historically it is derived from eolian (windblown) sandseas

(dunes) which developed inland, eastward of the middle Jurassic period Carmel Sea. It

has been subjected to a comprehensive geologic study [12,13,14]. The data investigated

here were taken from an outcrop approximately 36 m high, with a base area of roughly

0.1 km2 near the Glen Canyon Dam close to the Arizona { Utah border. Characteristic

features of Page Sandstone include:

{ laterally extensive, cross{bedded sandstone facies, primarily composed of �ne to very{

�ne grained, moderately well sorted, subrounded, positively{skewed, mesokurtic to

very leptokurtic, white, gray{white and yellow{orange quartzose clasts,

{ numerous thick (1 to 18 m), tabular{planar and wedge{planar, high{angle (14 to

31�), concave{upward cross{strata consistently oriented in a south{south{westerly

direction with extensive, marginally{located, horizontally{laminated, wind at strata

(sand sheets) and interdune sands appearing as the dominant structures throughout

the sandstone facies, and

{ wind{formed ripples, horizontal bedding (�rst{order) surfaces, deformed cross{strata

(sand slumps), frosted grains and wind de ation lag gravel appearing as associated

bedding structures and textures.

Extensive permeability data, collected with a mechanically{based �eld permeameter were

taken on core and surface samples. We consider only the core data, which is the most

extensive of the Page sandstone sets, and comprises a well{instrumented, one dimensional

data set. It has the additional feature of not having been exposed to weathering.

Core data. Permeability measurements were taken at an average spacing of 1.27 cm on a

40 m vertical core (more than 2800 points) extracted from the outcrop. Two measurements

5

were taken at each point, oriented at 90� to each other in the horizontal plane. They are

referred to as the CTRAN1 and CTRAN2 data sets.

0 5 10 15 20 25

permeability (Darcies)

-40

-35

-30

-25

-20

-15

-10

-5

0de

pth

(met

ers)

E14

E12

E11

E7

D2/D8

C13

C11

C8I3

C3

B-Complex

A9/A10/A11

Unit 1

Unit 2

Unit 3

PAG

E U

NIT

SN

AV

AJO

Figure 3.1: Closely spaced permeability measurements from a vertical core through thePage Sandstone outcrop. Referred to as the CTRAN1 data set in the text. (After Figure3.25 of reference [12].)

Figure 3.1 presents the CTRAN1 data set. Included on the right hand{side of the

plot are the locations of the �rst order surfaces separating the major cross{bedded strata.

Note that the data contains about 7.5 meters of measurements from the underlying Navajo

sandstone unit. We include this data in our analysis as it appears to represent a perme-

ability structure similar to that of to the Page Sandstone. The region of depth marked I3

corresponds to an intruding tongue of the Carmel Sea formation consisting of non{eolian

marine deposits of extremely low permeability. These data from the I3 section is not

6

0 5 10 15 20 25

permeability (Darcies)

-40

-35

-30

-25

-20

-15

-10

-5

0

dept

h (m

eter

s)

Figure 3.2: Closely spaced permeability measurements from a vertical core through thePage Sandstone outcrop. Referred to as the CTRAN2 data set in the text. (After Figure3.26 of reference [12].)

included in our analysis.

Figure 3.2 presents the CTRAN2 data set. The CTRAN1 and CTRAN2 data sets are

remarkedly similar, indicating a fair degree of isotropy in the horizontal direction in the

Page Sandstone.

Analysis of the CTRAN data from quantile{quantile plots by [12] gives best indicator

p values for the power{normal model (1) of p = 0:569 for the CTRAN1 data and p = 0:526

for the CTRAN2 data. Figure 3.3 is a histogram plot of the resultant � data.

The two point correlation functions h�� ��i(r) for the CTRAN1 and CTRAN2 data

sets are shown in Figure 3.4. Superimposed as a dashed line on each correlation function

7

-2 0 2 4 6 8

(K - 1)/0.569

0

50

100

150

200

250

300

0.569-2 0 2 4 6 8

(K - 1)/0.526

0

50

100

150

200

250

300

0.526

Figure 3.3: Histogram of the CTRAN1 (left plot) and CTRAN2 (right plot) data underthe indicated power{law transformations.

0 5 10 15 20

r (meters)

-2

-1

0

1

2

corr

elatio

n

CTRAN1

0 5 10 15 20

r (meters)

-2

-1

0

1

2

corr

elatio

n

CTRAN2

Figure 3.4: h�� ��i(r) correlation function for the CTRAN1 and CTRAN2 data sets overthe full data distance. The dotted curves indicate estimates of error due to sampling size.

trace is an estimate of the size of error due to loss of sampling statistics with distance. We

estimate the variance of the sampling error to be

var(h�i(r)) � (h�i(0))2an(r)

; (8)

where n(r) is the number of independent samples at distance r. We estimate n(r) as

8

n(r) = Nl=l, where Nl is the number of correlation samples at lag distance r = l �r, �r

being the constant spacing separating permeability values used in the correlation analysis.

In our analyses, �r is set to the average permeability spacing in the data set, and, where

necessary, linear interpolation is used to obtain permeability values at the required equally

spaced points.

We consider only the portion of the correlation function that lies continuously above

this error estimate. The resultant distance{truncated correlation plots are shown in Figure

3.5(a-c) as (a) linear{linear, (b) log{linear, and (c) log{log plots.

0.0 0.5 1.0

r (meters)

0

1

2

corr

elat

ion

(a)

CTRAN1

CTRAN2

0.0 0.5 1.0

r (meters)

1

corr

elat

ion

(b)

CTRAN1

CTRAN2

0.01 0.1 1

r (meters)

1

corr

elat

ion

(c)

CTRAN1

CTRAN2

0.01 0.1 1

r (meters)

0.0

0.5

1.0

β

(d)

CTRAN1

CTRAN2

Figure 3.5: Correlation plots truncated at a distance rmax above which sampling errorbecomes signi�cant: (a) linear{linear scaling, (b) log{linear scaling, (c) log{log scaling, (d)�(log(r)).

The correlation functions of the two data sets are very similar, in agreement with the above

remarked isotropy in the horizontal plane. The log{linear plot determines the applicability

of the commonly used exponential model

h�� ��i = �0e�r=� ;

9

having a single correlation length scale �. The application of multi{fractal model of x2 is

summarized in Figure 3.5(c) and (d). Figure 3.5(d) is obtained by numerical di�erentiation

of the data in Figure 3.5(c). The data was smoothed before di�erentiation to suppress noise.

The data is consistent with 0 � � < 1 at small distances. Although the data deteriorates

as r ! 1, it indicates growth of � consistent with expected loss of correlated behavior.

Figure 3.5(b) would indicate that for this bedded sandstone data, the exponential model

is not as good a model as it is for more homogeneous data (see following section).

4. Sand/Gravel Aquifer Data. Data sets from two sand and gravel aquifer �eld sites

were used: a sand quarry at Canadian Forces Base Borden, Ontario [8,9] and an abandoned

gravel pit near Otis Air Base on western Cape Cod [10,11].

Borden. The Borden data comes from an unconsolidated deposit (sand quarry) containing

a shallow water table aquifer. Two permeability (hydraulic conductivity) data sets are

available from 32 cores of aquifer material taken along two coring lines, one parallel to the

direction of mean groundwater ow (data set `AA'), the other transverse to the mean ow

(data set `BB'). The AA data set consisted of 20 cores, each spaced one meter horizontally

apart; the BB data set consisted of 13 cores, with the same horizontal spacing. The two

data sets form a `+', with core number 10 of AA and core 7 of BB being the intersection

between the two coring lines. Each core was approximately 2 meters in length, obtained

from depths between 2.5 and 4.5 meters below ground surface. Visual inspection of the

cores indicate [8] \the presence of numerous, discontinuous, strata (lenses) of coarse to silty

�ne{grained sand, embedded in a �ne{ to medium{grained sand. The contact between

lenses having a large textural contrast was usually sharp, and near{horizontal across the

cores. The thickness of individual beds varied from a few centimeters to a few tens of

centimeters, with the material within each bed being relatively homogeneous in texture,

although �ne laminations on the order of a millimeter to a few millimeters thickness were

sometimes encountered."

Conductivity measurements were taken on 5 cm length successive subsamples of each

core. As the measurements were taken at 22� C, the �eld value (at 10� C) of hydraulic

conductivity should be a factor of 1.36 less to account for viscosity and density corrections.

10

This correction factor is irrelevant to the analyses performed here.

Histogram and correlation function analyses from [8] applied in the vertical and hori-

zontal direction on both data sets indicate that the hydraulic conductivity is fairly isotropic.

Based on this conclusion, we ensemble average the data over the cores in each coring line,

compressing the data into two \vertical" samples, denoted `AA' and `BB'. In [8], a �{

squared goodness{of{�t test for the hypothesis that the hydraulic conductivity obeyed a

log{normal population distribution, as opposed to a normal distribution, resulted in ac-

ceptance with the probability of mistakenly rejecting the competing normal hypothesis

being less than 5 percent. As in [8], we thus assume a log{normal distribution for the

conductivities. The resultant log{normal histograms for the two data sets are presented

in Figure 4.1.

0.0001 0.001 0.01 0.1

K

0

20

40

60

0.0001 0.001 0.01 0.1

K

0

20

40

60

Figure 4.1: log(K) histograms of the Borden data sets: coring line AA (left plot) andcoring line BB (right plot).

The two point correlation function

h�� ��i(r) ; � � log(K)

for each data set is shown in Figure 4.2 (a). The correlation functions have been truncated

at r < 0:4 meters, based on the estimate of sampling error (8) applied to this data. In

[8] an exponential model for the correlation function was applied to the data. The model

appears to be a good predictor of the data; a �t results in

h�� ��i(r) = 0:75 exp(�r=�) ; � =

(0:12m vertical direction ;

2:8m horizontal direction :(9)

11

0.0 0.1 0.2 0.3 0.4

r (m)

0.00

0.04

0.08

corr

elatio

n

(a)

0.0 0.1 0.2 0.3 0.4

r (m)

0.00

0.04

0.08

0.0 0.1 0.2 0.3 0.4

r (m)

0.01

log(

corr

elatio

n)

(b)

0.0 0.1 0.2 0.3 0.4

r (m)

0.01

0.1

log(r)

0.01

log(

corr

elatio

n)

(c)

0.1

log(r)

0.01

0.1

log(r)

0.0

0.5

1.0

β

(AA)

(d)0.1

log(r)

0.0

0.5

1.0

(BB)

Figure 4.2: Correlation function analyses for the Borden AA (left) and BB (right) datasets. (a)h�� ��i versus r, (b) log(h�i) versus r, (c) log(h�i) versus log(r), (d) � versus log(r),

For comparison with our analysis (presented below), the predictions of this model are

shown in Figure 4.2(b) which presents the best straight line �t to the log(h�i(r)) data.

(Note that our normalization of the correlation function di�ers from reference [8]. It

is therefore not possible to read the slope and intercept presented in (9) directly from

Figure 4.2(b).) Applying the analysis presented in x2, the log{log plots of correlation are

presented in Figure 4.2(c) and the resultant distance dependent slope �(r) is presented in

Figure 4.2(d). Again the prediction is consistent with positive � � 1 at short distances.

Cape Cod. The Cape Cod tracer test site is situated in an abandoned gravel pit which

is part of a large sand and gravel outwash plain deposited during the retreat of continental

ice sheets about 12,000 years ago. The aquifer at the site is composed of about 100 m of

12

unconsolidated sediments that overlie a relatively impermeable, crystalline bedrock. The

upper 30 m of the aquifer, from which hydraulic conductivity measurements were taken,

consists of a permeable, strati�ed, sand and gravel outwash. The median (by weight)

grain size is about 0.5 mm, the outwash generally containing less than 1% silt and clay.

Hydraulic conductivity measurements were taken from permeameter measurements on 16

drill cores (labeled A through Q, with no core labeled `O'). The relative locations of the

core sites are shown in Figure 4.3.

14 18 22

distance (m)

80

90

100

110

120

dista

nce (

m)

Figure 4.3: Relative location of core sites at Cape Cod.

4

9

14

altitu

de (m

)

A B C D E F G H

4

9

14

altitu

de (m

)

I J K

0.00 0.10

K

L M N P Q

Figure 4.4: Hydraulic conductivity versus altitude for each of the Cape Cod cores.

Hydraulic conductivity versus altitude for each core is summarized in Figure 4.4.

13

These core logs indicate a variation in hydraulic conductivity of less than one order of

magnitude, resulting from interbedded lenses and layers of sands and gravels (supported

by visual examination of surface exposures).

-3 -2 -1 0

log(K)

0

25

50

75

(a)

0 1

r (m)

0.00

0.01

0.02

corr

elat

ion

(b)

0 1

r (m)

0.001

0.01

corr

elat

ion

(c)

0 1

r (m)

0.001

0.01

corr

elat

ion

(d)

0 1

r (m)

0.0

0.5

1.0

β

(e)

Figure 4.5: Histogram (a) and correlation function analysis (b-e) for the Cape Cod dataset: (b) h�� ��i versus vertical separation r, (c) log(h�i) versus r, (d) log(h�i) versus log(r),(e) � versus log(r).

Again, as this sample is relatively isotropic, we ensemble average over all cores in

the analysis presented here. The hydraulic conductivity is well modeled by a log{normal

distribution, as shown in Figure 4.5 (a). The correlation function analysis is summarized

in Figure 4.5 (b-e). The correlation, h�� ��i, � � log(K), versus vertical separation r is

displayed in Figure 4.5(b). Also shown (dashed line) is our estimate (8) of sampling size

error. As above, we truncate the correlation function plot when the trace falls below the

estimate of sampling size error, which occurs here for r � 0:9 m. The log{linear plot of

Figure 4.5(c) indicates that the exponential model is again a good descriptor of the short

distance behavior in this relatively homogeneous data. The log{log correlation plot is

14

shown in Figure 4.5(d) and the measured slope �(log(r)) in Figure 4.5(e). The more rapid

growth of � indicates a more rapid loss of correlation with distance than in the Borden

data.

5. Wag6 Data. The data sets used above correspond to intensive data collections, at

small spacings, on relatively homogeneous, well characterized sites. We turn our attention

to a \more typical" data set, taken from a site referred to as Wag6, on the Oak Ridge

Reservation (ORR), the site of the Oak Ridge National Laboratory. A history of study of

the ORR hydrology is summarized in [15]. The ORR consists of two broad geologic units,

the Knox aquifer and aquitards. As is the case for Wag6, the aquitards, where most of the

laboratory's waste sites are located, are mostly fractured shale, siltstone, sandstone, and

thinly bedded limestone units, characterized by low to very low permeability that decreases

rapidly with depth. The most pervasive structural feature is extensional, hybrid, and shear

fractures from the faulting and folding character of the region. It is indeed quite clear that

any data set accumulated on small scale, relatively homogeneous sites, are inadequate for

a description of the ORR.

23000 24000 25000 26000

distance east (feet)

15000

16000

17000

18000

dist

ance

nor

th (f

eet)

T wellsHHMS wellsETF wells

Figure 5.1: Relative areal location of the core samples comprising the Wag6 data set.

Coupled with the extremely heterogeneous geology of the site, the permeability data

sample is quite limited. The sample for the Wag6 site consists of 100 single permeability

measurements for 100 di�erent wells from di�erent locations and screened at di�erent

depths. Figure 5.1 presents the relative areal locations of the cored sites. The elevations

of the permeability measurements are displayed in Figure 5.2.

15

15000 16000 17000 18000

distance north (feet)

400

500

600

700

800

eleva

tion

(feet

) T wellsHHMS wellsETF wellssurfacewater level

23000 24000 25000 26000

distance east (feet)

400

500

600

700

800

eleva

tion

(feet

) T wellsHHMS wellsETF wellssurfacewater level

Figure 5.2: Altitudes of the permeability measurements in the Wag6 data set shown onnorth{south and east-west projections.

This data set di�ers in that the permeability values are obtained largely from slug

tests and are thus averaged over larger volumes than the permeameter derived data of

the previous sets. The data set is not very satisfactory for correlation function analysis;

it is clear that such a \typical" data set can only be dealt with in such a manner as

to attempt to determine average properties. In keeping with our basic assumption, that

geologic properties be slowly varying in logarithmic variables, we investigate the average

dependence of log{permeability with depth. The data shows a general drop in measured

16

permeability with depth. As shown in Figure 5.3, there is indication of an average linear

relationship between mean log permeability and depth, i.e. a mean permeability with a

nonstationary depth dependence of the form

�K(z) = K0e�z=45 ; with K0 = 2 � 10�4: (10)

0 100 200 300 400

depth (feet)

-9

-6

-3

log(K

(cm/

sec))

Figure 5.3: Wag6 data; log(K) versus depth.

A correlation analysis was performed on this data set by assuming a log{normally

distributed permeability. As the correlation calculation requires data equally spaced in

depth, z, linear interpolation over an average lag spacing was used. As only a few, widely

spaced points are found at the lower depths, their inclusion in the correlation function

would lead to an overly correlated estimate. In order to avoid this, the data set was

truncated to the interval 0 < z � 125 feet.

The analysis is summarized in Figure 5.4. The histogram of the � � log(K) data is

shown in 5.4(a).

In �gures (b-d), the z-dependent mean de�ned by (10) is subtracted from the � data.

The result is a short range of statistically signi�cant correlation, and a positive �. In

�gures (e-g), a global z-independent mean is subtracted. The result is a longer range of

statistically signi�cant correlation, but still the resulting � is positive. While method (b-d)

is preferable as a statistical analysis of data with signi�cant trends, the plots from (e-g)

show that (unsubtracted) trends do not (at least in this case) result in a � < 0.

17

-6 -4 -2

log(K)

0

2

4

6

(a)

0 2 4 6 8 10

r

0.0

0.2

0.4

corr

elat

ion

(b)

1 10

r

0.1

corr

elat

ion

(c)

1 10

log(r)

0

1

2

3

β

(d)

0 2 4 6 8 10

r

0.0

0.2

0.4

corr

elat

ion

(e)

1 10

r

0.1

corr

elat

ion

(f)

1 10

log(r)

0

1

2

3

β

(g)

Figure 5.4: Wag6 data set. (a) Histogram of � � log(K), (b-d): correlation analysis withdepth dependent mean subtracted, (e-g): correlation analysis with depth independentmean subtracted. (b and e) h�� ��i versus vertical separation r, (c and f) log(h�i versuslog(r), (d and g) � versus log(r).

6. Discussion and Conclusions.

1. Geological data for spatial variation of correlations is statistically signi�cant only over

a small range of length scales, even for small, relatively homogeneous, well instru-

mented sites.

2. True multi-length scale data from a single site will depend on use of di�erent (length

scale dependent) measurement methods to minimize noise. Many small scale mea-

surements, averaged or homogenized over a larger volume will give an e�ective perme-

ability for that larger volume in principle, but such measurements will be impractical

due to the large number required. In place of this, large length scale correlations will

18

have to use measurements which sample a large volume intrinsically. Methods for

the comparison and consistent interpretation of distinct measurement methods taken

from the same site is an open problem, and one that will have to be solved before

multi-length scale date is available and can be interpreted consistently.

3. Primitive variables expressed as a power law (e.g. K1=2) may be preferable to either

normal or log normal distributions. A one parameter family, Kp�1ap , of power normal

distributions interpolates between normal (p = 1) and log normal p = 0.

4. Fractal and multifractal distributions �t short distance data about as well as expo-

nential models.

5. Fractal and multifractal models �t multiscale long distance hydrological data, while

exponential models do not �t this data, especially for multilength scale data combined

from distinct sites.

6. Multifractal distributions appear to be more suitable than fractal ones for short as

well as long distance data. The range � 2 [0; 1] appears to �t short distance geological

data, as it does for long distance data hydrological data.

7. Nonstationary distributions can occur, as trends in real data. In the data set exam-

ined, for which trends were important, the correlation function resulted in positive �

(decaying correlations), regardless of whether the subtracted mean was depth depen-

dent or global, i.e. did or did not subtract the trend information.

References

[1] J. GLIMM and D. H. SHARP, A Random Field Model for Anomalous Di�usion in

Heterogeneous Porous Media, J. Stat. Phys., 62 (1991) 415-424.

[2] Q. ZHANG, A Multi-Length Scale Theory of the Anomalous Mixing Length Growth

for Tracer Flow in Heterogeneous Porous Media, Journal of Statistical Physics, 66,

(1991), 485{501.

[3] F. FURTADO, J. GLIMM, W. B. LINDQUIST, and F. PEREIRA,Multi Length Scale

Calculations of Mixing Length Growth in Tracer Floods, Proceedings of the Emerging

Technologies Conference, Houston TX, pp. 251{259. F. Kovarik, ed., Institute for

Improved Oil Recovery, U. Houston, 1990.

19

[4] F. FURTADO, J. GLIMM,W. B. LINDQUIST, and F. PEREIRA, Characterization of

Mixing Length Growth for Flow in Heterogeneous Porous Media, SPE paper #21233,

1991.

[5] F. FURTADO, J. GLIMM, W. B. LINDQUIST, F. PEREIRA, and Q. ZHANG,

Time Dependent Anomalous Di�usion for Flow in Multi-fractal Porous Media, Lec-

ture Notes in Physics, 398, (1992) 79{89.

[6] J. GLIMM, W. B. LINDQUIST, F. PEREIRA and R. PEIERLS, The Multi-fractal

Hypothesis and Anomalous Di�usion, Revista Brasileira de Matematica Aplicada e

Computacional, 11, (192).

[7] S. P. NEUMAN, Universal scaling of hydraulic conductivities and dispersivities in

geologic media, Wat. Resour. Res., 26 (1990) 1749{1758.

[8] E. A. SUDICKY, A natural gradient experiment on solute transport in a sand aquifer:

Spatial variability of the hydraulic conductivity and its role in the dispersion process,

Water Resources Research, 22 (1986) 2069{2082.

[9] D. L. FREYBERG, A natural gradient experiment on solute transport in a sand

aquifer: 2. Spatial moments and the advection and dispersion of nonreactive trac-

ers, Water Resources Research, 22 (1986) 2031{2046.

[10] D. R. LEBLANC, S. P. GARABEDIAN, K. M. HESS, L. W. GELHAR, R. D.

QUADRI, K. G. STOLLENWERK, AND W. W. WOOD, Large{scale natural gradi-

ent tracer test in sand and gravel, Cape Cod, Massachusetts 1. Experimental design

and observed tracer movement, Water Resources Research, 27 (1991) 895{910.

[11] S. P. GARABEDIAN, D. R. LEBLANC, L. W. GELHAR, and M. A. CELIA, Large{

scale natural gradient tracer test in sand and gravel, Cape Cod, Massachusetts 2.

Analysis of spatial moments for a nonreactive tracer, Water Resources Research, 27

(1991) 911{924.

[12] D. J. GOGGIN, Geologically{sensible modeling of the spatial distribution of perme-

ability in eolian deposits: Page sandstone (Jurassic), northern Arizona, University of

Texas at Austin thesis, 1988.

[13] M. A. CHANDLER, Depositional controls on permeability in an eolian sandstone

sequence, Page Sandstone, Northern Arizona, MA Thesis, The University of Texas at

20

Austin, Dec. 1986.

[14] J. B. KNIGHT, Eolian bedform reconstruction { a case study from the Page Sandstone

(Juraissic), Northern Arizona, MA Thesis, The University of Texas at Austin, Dec.

1986.

[15] D. K. SOLOMON, G. K. MOORE, L. E. TORAN, R. B. DRIER, and W. M. MC-

MASTER, Status report: A hydrogeologic framework for the Oak Ridge Reservation,

ORNL/TM-12026 (1992).

21