Worthington,P.F.,1985 -The Evolution of the Shaly-Sand Concepts in Reservoir Evaluation

8/10/2019 Worthington,P.F.,1985 -The Evolution of the Shaly-Sand Concepts in Reservoir Evaluation

1/18

ABSTRACT

A wide variety of procedures are currently in routine

use for the evaluation of shaly sands. Each of these can

furnish a significantly different reservoir evaluation.

Yet, no one method predominates within the industry.

As

a

means of investigating this unsatisfactory state

of

affairs, the development of thinking about the shaly

sand problem has been mapped up to the present

day.

In

so

doing, attention has been focused on the

manifestation of shale effects in electrical data, since

this remains the most contentious area through its

bearing on the determination of water saturation and

thence hydrocarbons in place. By considering the basic

characteristics of the underlying petrophysical models,

it has become apparent that the multifarious equations

for determining water saturation from electrical data

can be ordered into type groups. Thus, seemingly

dissimilar models can be related from a formation-

evaluation standpoint. This subject area is therefore

more systematic than it might initially have appeared.

Using this type classification as a basis, an assess-

ment has been made of further developments in this

complex subject area. It is concluded that the shaly-

sand problem will only be truly solved from an elec-

trical standpoint when the requirement of a flexible,

representative algorithm, based on a sound scientific

model, which can be applied directly to wireline data,

has been fully met.

INTRODUCTION

This paper is based upon earlier technical presenta-

tions by the author to several chapters of the SPWLA.

The text has been prepared in response to requests for a

transcript of those presentations.

The object is to provide an insight into the origins of

some of the resistivity equations currently used for the

determination

of

water saturation S, in shaly sands, a

far reaching aspect of the shaly-sand problem and one

which remains controversial. This is attempted by

examining the growth of understanding from the

emergence of shaly-sand concepts through to the pre-

sent day. It is not proposed to advance a comprehensive

treatise on the

S ,

component of the shaly-sand prob-

lem, but rather to identify key stages in the develop-

ment of ideas. The treatment is therefore based very

much on a personal interpretation of this evolutionary

process; other petrophysicists would no doubt chart

these trends differently. Furthermore, it is not the inten-

tion to underwrite or refute a particular conceptual

model, but rather to seek an ordering of what might

appear, at first sight, to be an uncoordinated collection

of equations. Thus, the inclusion or omission of a par-

ticular method does not imply approval or disapproval,

respectively, of the technique in question.

Confronted at the outset with over

30

S , models

from which to choose, this treatment has been given

greater poignancy by focussing on those conceptual

models wherein the shale parameters are notionally

determinable from downhole wireline measurement.

Despite this self-imposed direction, it is hoped that this

appraisal will help to clarify some of the thinking that

underlies the application of the available

S,

options in

the evaluation of shaly reservoirs.

The words shale and clay are used synonymously

here. To draw upon the distinctions that have been

made elsewhere would not be helpful for present pur-

poses, since the stated objectives of this work require a

simplification of what is already an exceedingly com-

plex problem. Furthermore, primary emphasis is given

to dispersed shales, rather than laminated or structural

shales, since these have received by far the greatest

attention in the literature. The word shale will there-

fore relate to dispersed shalelclay unless otherwise

qualified.

EMERGENCE OF TH E SHALY-SAND PROBLEM

The period prior to 1950 can be seen as a shale-

free period from a petrophysical standpoint; it is only

since this date that the shaly-sand problem has been

fully recognized and addressed.

Selected petrophysical developments during the

shale-free period are itemized in Table

1.

Surface

resistivity prospecting, pioneered by Wenner in North

America and by the Schlumberger brothers in Europe,

was the precursor of geophysical well logging fifteen

years later. The development of the first quantitative

THE EVOLUTION OF SHALY-SAND

CONCEPTS IN RESERVOIR EVALUATION

PAUL F. WORTHINGTON

The British Petroleum Company

Sunbury-onThames, England

THE

LOG ANALYST 23


2/18

TABLE 1

CHRONOLOGICAL LA NDMARKS OF T HE

SHALE FREE PERIOD

PARTIALLY EXTRACTED

FROM

JOHNSON,

1961)

1812

1869

1883

1912

1927

1932

1939

1942

1947

1947

1948

1949

Electrical phenom ena measured in the walls

of Cornish tin mines

Downhole temperature measurements

In-situ determination of rock resistivity by

measurem ent at the earths surface

Resistivity prospecting established

First electric log

Quantitative resistivity tool Norm al device)

Natural gam ma tool*

Archies laws

Induction log

Recognition of in terface conductivity in

reservoir roc ks*

Determination of R, from the SP

log

Appreciation of SP response in shaly

sands*

*denot es shale-related developm ent

resistivity tool, the Nor mal device, an d th e publication

of

Archies empirical laws ten years afterwards, pro-

vided the basis for the quantitative petrophysical

evaluation

of

arenaceous reservoirs. Although Archies

laws were established specifically for clean sands, the

increasing num ber of sha le related developments dur-

ing the last 10-12years

of

the shale-free period is indica-

tive

of

a

growing awareness of the interpretative com -

plexities associated with the shaly-sand problem.

The emergence

of

the shaly-sand problem as it

affects resistivity data can be more readily traced by

considering only conditions

of

full water saturation in

the first instance. A convenient starting point

is

the

definition of formation factor

F

which was the first of

three eq uation s propose d by A rchie (194 2), viz.

where

R,

s th e resistivity

of

a reservoir rock whe n fully

saturated w ith aqu eous electrolyte of resistivity R,, and

C,

and C, are the corresponding conductivities. A plot

of C, vs C, for a given sample should furnish a straight

line of gradient

1

IF provided that Archies experi-

mental conditions

of a

clean reservoir rock fully sat-

urated with brine are completely satisfied. Subject to

these conditions th e forma tion factor is precisely what

the name implies; it is a parameter of the formation,

more specifically on e th at describes the pore geometry.

It is independent

of

C, so tha t a plot of C,/C. vs C,

for

a

given sample should furnish a straight line parallel

to the C, axis, Figure 1.

However, aroun d 1950 there w as increasing evidence

from various formations to suggest that the ratio C,/C,

is not always a constant for a given sample but can

actually decrease

as

C, decreases (Patnode an d Wyllie,

1950).T he relative decrease in C,/C, a t a given level

of

C, appeared to be more pronounced for shalier

specimens (Fig

1).

Since C, was presumed to be known,

the on ly possible explanation for this phenom enon

lay

in the effect of the shale component of the reservoir

rock upon C,. This effect was essentially to under

reduce C, as C, decreased or, to put it anoth er way, to

impart an extra conductivity to the system at lower

values of C,.

For

this reason th e electrical manifesta-

tion of shale effects has been described in terms of an

excess conductivity (Winsauer and McCardell,

1953). It becam e advisable to regard the ratio C,/C,

as

an apparent formation factor F, which is equal to the

intrinsic formation factor

F

only when Archies

assumptions are satisfied. Throughout this paper the

symbol F identifies the formation factor as defined by

Archie while F, represents no more than a salinity-

dependent approximation thereto.

Since the Archie definition of equation 1) was not

found to be valid for all formations, a more general rela-

tionship between C, and C, was sought in order to

accommodate the excess conductivity.

By

rewriting

equation

(1)

as

C,

c, =

and incorporating the excess conductivity within a

composite shale-conductivity term X, it was proposed

that an expression

of

the form

(3)

is valid for all granular reservoirs that are fully water

saturated.

For a clean sand,

X - 0

and equation (3) reduces to

equation

(2).

If C, is very large, X has comparatively

Clean sand

I

I

c w -

Figure

1 Schematic variation of th e rat io C,/C,, =F a)

with C, for shaly san ds .

24 JANUARY-FEBRUARY,

1985


3/18

little influence on C, and again equation (3) effectively

reduces to the Archie definition. Conversely, the ratio

C,/C, is effectively equal to the intrinsic formation fac-

tor

F

only if X s sufficiently small andlor C, is suffi-

ciently large. Thus, although the absolute value of X

can be seen as an electrical parameter of shaliness, the

manifestation of shale effects from an electrical stand-

point is also controlled by the value of X relative to the

term C,/E

During the period 1950-1955 evidence began to

accumulate that the absolute value of the quantity

X

is not always a constant for a given sample over the

experimentally attainable range of C,, as equation (3)

would appear to imply, but can vary with electrolyte

conductivity (Winsauer and McCardell, 1953: Wyllie

and Southwick, 1954: Sauer et al., 1955). The most

widely accepted behavioural pattern, which has con-

tinued to be supported (Waxman and Smits, 1968;

Clavier, Coates and Dumanoir, 1977, 1984),was that for

a given sample, the absolute value of X ncreases with

C, to some plateau level and then remains constant as

C, is increased still further. This pattern is illustrated

for hypothetical data through Figure 2. Here the terms

non-linear zone and linear zone have been adopted

for the regions of variable X and constant

X,

respectively.

The implications of changes in C,, and thence in the

relative (but not necessarily the absolute) value of

X,

are illustrated in Figure 3 for the formation factor vs

porosity relationship. Data are from the Triassic Sher-

wood Sandstone of northwest England. For each core

sample the value of X s known to be constant over the

particular range of C, represented here. Figure 3a

depicts a plot of intrinsic formation factor

F

vs porosity

9 for conditions corresponding to a high C, and

thereby a relatively insignificant

X.

The

data

distribu-

6

a

a

1 -

t

C O

0

0

1

I I

N o n

-

l i n e a r

z o n e

Linear

z o n e

Figure 2 Schematic variation

of C,

with C, for water-

saturated shaly sands.

tion supports the generalized form of the formation fac-

tor-porosity relationship, a variation of the second

equation proposed by Archie (1942),viz.

a

9

F =--

(4)

where a and m have usually been assumed constant for

a given reservoir. In contrast, Figure 3b relates to condi-

tions of sufficiently low C, that the same absolute

values of X epresented in Figure 3a have now become

highly significant even though the sample population

remains unchanged. Because of this, the ratio C,/C,

now represents merely an apparent formation factor Fa.

These departures from the Archie assumptions result in

a

breakdown of the linear trend of Figure 3a to such a

degree that there is no longer a useful relationship.

0

Ilr

-

4

10

15

20

P o r o s i t y 1

0

30

Im

P o r o s i t y Io/ol

Figure 3

Comparison of a)

F

vs 4 and b) Favs 6

crossplots for 19 sandston e samples dat a

from Worthington and Barker, 1972).

THE

LOG ANAL YST

25


4/18

Although Figure 3 contrasts extreme cases, the

implications of this disparity a re

of

general significance.

Prior to the development of reliable porosity tools it

was often the practice to estimate

4

from the ratio

C, /C , using a standard version of equation

(4)

n con-

junction with resistivity logging data from nearby

water zones. In

so

doing it was essential to have suffi-

ciently clean c onditions for there t o be a well defined

relationship between

C#I

and C, /C, . Where this condi-

tion was satisfied it was still possible to proceed even if

the ratio C,/C, actually represented an apparen t forma-

tion factor Fa instead of the intrinsic formation factor

In

the former case a and

m

would be pseudo-

parameters which would compensate for any departure

of Fa

from

F

when calculating porosity. This approach

required that the input value

of

F, related to the same

C, as tha t used to establish the relationship between

F,

an d in the first place.

Th e advent of porosity tools has resulted in a change

of

usage

of

equation

(4).

t is current practice to infer

4 from porosity tool response(s) and then to calculate

F

using pre-determined values

of a

and

m.

n this case

the resulting value of F will be wrong if the param eters

a

and

m

do not themselves relate specifically to effec-

tively clean conditions but have inadvertently been

established

on

the basis of a correlation of

F.

with

4 .

This error can be readily transmitted to subsequent esti-

mates

of

water saturation.

The development of suites of porosity tools has

brought with it an e ntirely different aspect of the shaley-

sand problem, that

of

correcting radiometric and sonic

tool responses for shale fraction. H owever, the resulting

shale corrections for porosity have rarely been as con-

tentious as the various procedures adopted in the

ongoing quest for reliable shale-corrected water satu-

rations. Therefore althoug h u ncertainties in log-derived

porosities are capable of in ducing

a

significant error in

subsequently estimated values of S,, the porosity com-

ponent of the shaly-sand problem is not considered

further here even thoug h it remains a potentially diffi-

cult area. Instead, attention will be concentrated on the

S,

problem much

of

which is concerned with the

physical significance of the quantity X n equation (3).

For the time being these considerations will continu e to

be restricted to water zone conditions for, as implied

earlier, the premature inclusion of an S, term would

unnecessarily complicate the treatment of what

is

already a very complex problem.

EARLY SWALY-SAND CQNCEPYlS

Prior to 1950 it had been the convention to regard a

water saturated reservoir rock as com prising two com -

ponents, a nonconducting matrix and an electrolyte.

Where these specifications were satisfied the ratio

C,/C, was not a functio n of C, a nd th e Archie condi-

tions were met. In other cases, these simple specifica-

tions were inadequate to account for the excess

conductivity phenomenon which gave rise to a non-

trivial value of

X

in equation (3).

In a n attemp t to acco unt for this electrical manisfesta-

tion

of

shale effects Patnode a nd Wyllie (1950)proposed a

two-element, shalysand model comprising conductive

solids and an electrolyte. In this case the quantity

X

was described

as

the conductivity due t o th e conduc-

tive solids

as

distributed in the core. Since

X

was

found t o be a co nstant over the range of C , considered,

this m odel can be identified retrospectively with the so

called linear zone of Figure

2.

L. de Witte (1950) observed that the Patnode-Wyllie

model is equivalent to two parallel resistances, one

representing the resistance of the water phase and the

other equal to the total resistance of the conductive

solids as distributed. He argued that this would

require th e electrolyte and conductive solids to be elec-

trically insulated from one another; since they were

not, th e Patnode-Wyllie model w as untenable. L. de

Witte did, however, draw upon Patnode and Wyllies

work on clay slurries to propose that a homogeneous

mixture of conductive solids and electrolyte behaves

exactly as a mixture of two electrolytes. The resulting

two-element, conceptual model comprised a non-

conducting matrix and a clay slurry electrolyte.

Because one of these elements is actually

a

composite

system, the co rresponding resistivity algorithm does not

conform to the generalized equation (3).However, it is

of

linear form an d therefore describes the linear zone

of

Figure 2.

Winsauer a nd M cCardell (1953) ascribed th e abn or-

mal conductivity of shaly reservoir rocks to the elec-

trical double layer in the solution adjacent to charged

clay surfaces. This abnormal, or excess, double-layer

conductivity was attributed to adsorption on the clay

surface and a resultant concentration of ions adjacent

to this surface. The Winsauer-McCardell model

takes

the form of equa tion (3) with X = z/ F where

z

is the

excess double-layer conductivity. Thus the same

geometric factor F was supposed for both th e free elec-

trolyte an d th e double-layer comp onents of the parallel

resistor model. Furthermore the quantity z was shown

to vary with C,. T he variability of X n the non-linear

zone of Figure 2 was therefore accomm odated but little

evidence was presented for the constancy of

X

in the

linear zone. Nevertheless by proposing

a

variable

X

he

Winsauer-McCardell model differed fundamentally

from the earlier linear representations.

In order to accoun t for the non-linear zone of Figure

2 without having to postulate

a

variable shale-con-

ductivity-term Wyllie and Southwick (1954) extended

th e Patnode-Wyllie model to a three elem ent system.

This comprised conductive solids and electrolyte com-

pone nts as before, with

a

third component consisting of

electrolyte and conductive solids arranged in series.

26

JANUARY-FEBRUARY, 1985


5/18

This additional component admitted some electrical

interaction between the solid and liquid phases. This

three element model gave rise to an additional interac-

tive term on the right hand side of equation (3).The

quantity

X

was set equal to the intrinsic conductivity

of the solid phase qualified by a n appropriate geo-

metrical factor. In this way the Wyllie-Southwick model

could be used to represent both the linear and non-

linear zones of Figure

2

without having to vary X.

L.

de

Witte (1955) formulated the concept of astrongly

reduced activity of the double layer counterions present

in a shaly sand. This resulted in a two element model

in which the total rock conductivity was taken

to

be the

sum

of

conductivity terms associated with the double

layer and with the free (or far) water. The model was

represented by a linear relationship and therefore

described only the linear zone of Figure

2.

Yet the

development is an interesting one for it lends support

to the Winsauer-McCardell model as

a

conceptual

forerunner of the contemporary double-layer models.

Hill and Milburn (1956) showed that the effect of

clay minerals upon the electrical properties of

a

reser-

voir rock is related

to

its cation exchange capacity per

unit pore volume QY . The measurement of

y

therefore

provided an independent chemical method

of

determin-

ing the effective clay content. Hill and Milburn

developed an exponential equation to relate C, to C,;

in sodoing, it was presumed that when C, = 100 Sm-

for a fully saturated reservoir rock, the electrolyte con-

ductivity is sufficiently high to suppress any shale

effects. This equation contained a b-factor which was

empirically related

to QY

and which was constant for a

given lithology. Thus, although the Hill-Milburn equa-

tion did not conform to the generalized equation

(3),

the shape

of

the C.-C, curve in Figure

3

was approxi-

mately represented through the use of an exponential

function without having to suppose variations in the

shale term, b. However, a major drawback of this

approach is that the C, function passes through a

minimum at some small value of C,. The model

therefore predicts that C, would increase as C, is

decreased below this value. It is physically untenable

that C, should increase as

C,

decreases, and it

is

prob-

ably for this reason that the Hill and Milburn method

was not taken further.

A. J. de Witte (1957) observed that Hill and Milburn

were unnecessarily complicating the issue, since their

data

could be equally well represented by equation

(3),

if one made allowance for some irregularity of the

plotted points. A. J. de Witte defined the product X F

as the shaliness of

a

reservoir rock, a composite

parameter which was independent of C,. Thus it was

the linear zone

of

Figure 2 that was being represented

by A. J. de Wittes model.

It should be noted that all these early models can be

described by equations which reduce to equation (2)

when the shale-conductivityparameter is insignificant.

This is true even for those models which cannot be rep-

resented by equation

(3).

At this point we can close the discussion of early

shaly-sand concepts. Despite the considerable attention

given to the shaly-sand problem during the 1950s, the

models described above collectively suffered from one

fundamental drawback n no case could the shale

related parameter be determined directly from logging

data.

Efforts therefore continued to be directed towards

finding a conceptual model which did not suffer from

this shortcoming.

CONTEMPORARYSHALYSAND CONCEPXS

For the purposes of this discussion the shaly-sand

models introduced since about 1960 have been divided

into two groups.

(i) Concepts based on the shale volume fraction, V s h .

These models have the disadvantage of being scien-

tifically inexact with the result that they are open

to misunderstanding and misuse. On the other

hand they are at least notionally applicable to log-

ging data without the encumbrance of a core

sample calibration of the shale related parameter.

(ii) Concepts based on the ionic double-layer phe-

nomenon. These models have

a more attractive

scientific pedigree. If strictly applied, they require

core-sample calibration

of

the shale related param-

eter against some log derivable petrophysical quan-

tity. Otherwise their field application might involve

approximations which effectively reduce the shale

term to one in v s h .

Although

V,

models are being progressively dis-

placed by models of the second group, this process will

not be complete until there exists an established pro-

cedure for the downhole measurement of X.

V s h

Models

The quantity v s h is defined as the volume of wetted

shale per unit volume of reservoir rock. This definition

takes account of chemically bound waters; in this

respect, it is analogous to that of total porosity.

v s h models gained credence because of earlier ex-

perimental work which showed potentially useful rela-

tionships between the amount of conductive solids

present within a saturated granular system, such as a

clay slurry, and the conductivity of the solid phase (e.g.

Patnode and Wyllie, 1950). These early

data

did not

relate to typical reservoir rocks and they have subse-

quently been extrapolated far beyond their original

limits. Attempts to explain the physical significance of

the parameterX n terms of v s h have had either a con-

ceptual or an empirical basis.

Hossin (1960) approached the problem from a con-

ceptual standpoint. The development can be traced by

drawing upon the aforementioned analogy between v s h

THE LOG ANALYST

27

1

_


6/18

.

and total porosity. It involves specifying a clean, fully-

saturated , granular system which satisfies the original

form of Archies law, viz. equ atio n (4) with

a = 1

and

m

=

2. With these specifications the eq uation can be

rewritten in the form

c, =

42

c, 5 )

Suppose now that the interstitial electrolyte is pro-

gressively displaced by w etted shale. Whe n this process

is complete th e volume that was previously pore space

is now th e volume of shale. Th us

4

is analogous to V s h .

Furthermore the conductivity

of

the material occupy-

ing this volume has changed from C, to wetted shale

conductivity csh. The term @C, of equation 5 ) is

therefore analogous to

V,: C s h .

The quantity C, is now

equivalent to

X

since there

is

no free electrolyte in th e

system. Thus

Whe re both shale and electrolyte are present, equation

(6) defines th e shale-related term

of

equation (3). Note,

however, that in these intermediate cases, the porosity

of the system is an effective porosity since the

chemically bound waters are included within

v&.

ince

there is no provision for X o vary with C,, the Hossin

model relates specifically to the linear zone of Figure

2.

The Hossin equation and other V,h relationships are

listed in Table

2.

= v s :

Csh (6)

TABLE 2

SHALY-SAND RELATIONSHIPS INVOLVING V,,

(WATER ZONE)

Hossin (1960)

Simandoux (1963)

Doll (unpublished)

Poupon and Leveaux

(1971)

Simandoux (1963) reported experiments on homo-

geneous m ixtures of sand an d montmorillonite. He pro-

posed an expression of the form

of

equation

(3)

with

the quantity X represented as th e product v , h Csh. This

equation (Table 2) also relates specifically to the linear

zone of Figure 2. The

V,h

term in the Simandoux equa-

tion does not strictly correspond to the wetted shale

fraction of the Hossin concept, since the natural

calcium montmorillonite used by Simandoux was not

in the fully wetted state when the mixtures were made.

Subject to this qualification, the Simandoux and

Hossin equations differ only in the exponent

of

v s h .

T he linear form of th e Hossin and Siman doux equa-

tions means that they provide only a partial representa-

tion

of

the behavioural pattern of Figure 2. They do

not represent data from the non-linear zone. However,

a V,, equation which does admit non-linear trends on

a C, vs C, plot is that ascribed to Doll (unpublished ) by

various authors (e.g., Desbrandes, 1968; Raiga-

Clem encea u, 1976). Because of th e lack of pub lished

documentation, the precise reasoning behind the Doll

equation remains unspecified. However, it can be seen

from Table

2

that the Doll equation can be written

down by separately taking the square root of each term

of

the Hossin equation. W hether this was the intention

is unclear, but the effect is to impart a non-linearity

which might allow the equation to be used for data

from the non-linear zone of Figure

2.

Furthermore, by

squaring the Doll equation we have

Equ ation (7) is partly of the form of the generalized

parallel resistor equation (3), but there is an addi-

tional, interactive term on the right hand side which

can be seen as representing any cross linkage between

the electrolyte and shale components. Interestingly,

Poupon a nd Leveaux (1971) noted that Doll did indeed

suggest such a cross-linkage term some 20 years ago.

T he published work of th at period (Wyllie and

Southw ick, 1954) furnished an equation

of

the form

where a, b and

c

are geometric factors. There is an

obvious correspondence between the terms of equa-

tions (7) and (8).Neither equation allows any variation

in the parameters

of

shaliness w ith C, .

Poupon and Leveaux (1971) proposed the

so

called

Indonesia form ula (Table 2), an expression which is

similar to the Doll equation but with V , h having an

exponent that is itself

a

function of

V s h .

This equation

was developed for use in Indonesia because there com-

paratively fresh formation waters and high degrees of

shaliness had exposed the shortcomings of other eq ua-

tions. It has subsequently found an application else-

where. As with the Doll equation , the Poupon-Leveaux

relationship accommodates the non-linear zone of

Figure 2.

It is worth emphasizing that none

of

the four equa-

tions of Table

2

allows a complete representation of

rock conductivity data over the experimentally attain-

able range

of C,.

Figure

4

ompares the two types of

V,h equation considered here, the two and three ele-

ment, parallel resistor equations. The two element

equation can provide a reasonably correct data

representation only in the linear zone of Figure

2.

If it

is desired to improve the mismatch over part of the

28

JANUARY-FEBRUARY,

1985


7/18

non-linear zone, this can only be accomplished for a

given

ah

by decreasing c s k so that a mis-match is intro-

duced in the linear zone. Similarly, a good representa-

tion through a three element equation in the non-linear

zone can only be attained at the expense of a mis-

match in the linear zone. This mis-match can be

improved for a given sh y decreasing C,, but in

so

doing, the fairly accurate representation in the non-

linear zone must be partially sacrificed. Figure 4,

therefore, summarizes the physical implications of

enforced changes in

C,,,

which are made in order to

improve the consistency of a particular log evaluation

exercise.

t

C O

Linear

zone

o n-

l inear

I

cw -

Figure 4

Schematic com parison

of

C, C, dat a rep-

resentations

by

1) two element and

(2)

three

element

V

conductivity models.

Apart from the unavailability of a universal sh

equation there is one other major disadvantage of vsk

models; the v& arameter does not take account of the

mode of distribution or the composition of constituent

shales. Since variations in these factors can give rise to

markedly different shale effects for the same numerical

shale fraction, improved models were sought which did

take account of the geometry and electrochemistry of

mineral-electrolyte interfaces.

Double-LayerModels

The term double layer model is used here to

describe any conceptual model which draws directly or

indirectly upon the ionic double-layer phenomenon,

as

described for reservoir rocks by Winsauer and

McCardell (1953). In this respect, their work can be

seen as a conceptual forerunner of the models described

below, all of which furnish an expression of the same

general form as equation (3).

Waxman and Smits (1968)explained the physical sig-

nificance of the quantity X n terms of the composite

term

BQ,/F*,

where

Qv

is the cation exchange capacity

per unit pore volume,

B is

the equivalent conductance

of sodium clay exchange cations (expressed as a func-

tion of C, at 25C) and

F*

is the intrinsic formation

factor for a shaly sand, Table 3. The product BQy is

numerically equivalent to the excess conductivity

z

of

Winsauer and McCardell (1953). Thus, the Waxman-

Smits model also assumes that the conducting paths

through the free pore water and the counterions within

the ionic double layer are subject to the same geometric

factor

F*.

The dependence of

B

upon C, allowed

X

to

vary with

C , so

that both the non-linear and the linear

zones of Figure 2 could be represented through one

parallel-resistor equation.

Clavier et al. (1977, 1984) sought to modify the

Waxman-Smits equation to take account of experi-

mental evidence for the exclusion of anions from the

double layer. This was done in terms of a dual water

model of free (formation)water and bound (clay)water.

It was argued that a shaly formation behaves as though

it were clean, but with an electrolyte of conductivity

C,, that is a mixture of these two constituents. Thus

the Archie definition of equation (2) was rewritten

(9)

C,.

c,

=-

F o

where

F,

is the formation factor associated with the

entire pore space (i.e. both free and bound water). Equa-

tion (9) forms the basis of the dual water equation

(Table 3). It can be inferred fmm Table 3 by rearranging

the dual water equation that the geometric factors

associated with the two parallel conducting paths are

not equal. Furthermore, the presence of the variable

parameter vQ n the shale term allowed

X

o vary with

C, at low salinities. This meant that both the non-

linear and the linear zones of Figure 2 could be repre-

sented through this one equation.

TABLE 3

SHALY-SAND RELATIONSHIPS

FOR

WATER ZONE)

DOUBLE-LAYER MODELS

c , = - + -, BQ

F* F*

c s u 6

x2F

, = - +

F

Waxman & Smits

1968)

Rink &

Schopper 1974)

c, =c

(Cbw-CW)v,Q, Dual-water mo del:

Clav ier et al . 1977,

1984)

+

F F

It is important to note the distinction drawn between

F*

of the Waxman-Smits equation and

F,

of the dual

water model. For a clean sand F* = F., and both are

THE

LOG ANALYST 29


8/18

..

.

.

.

.

.

~

.

.~

equivalent t o the Archie formation factor F. For a shaly

sand F* is notionally the formation factor that the

reservoir rock w ould possess, if the solid clays were to

be replaced by geometrically identical but surficially

inert matrix, the bound water being grouped with the

free water as a uniform equivalent electrolyte. However,

Clavier et al. (1977, 1984) note t hat measured values of

F*, obtained from multiple salinity determinations of

rock conductivity, are affected by the presence of

bound water. The quantity F, is claimed to be an

idealized formation factor expressed as the product of

F* and a correction factor for th e geometrical effect

of

the bound water.

An important point of qualification is that the

Waxm an-Smits and dual water models are specifically

based on the cation exchange properties of sodium

clays in th e presence

of

an NaC l electrolyte as observed

for those reservoir rocks represented in the underlying

experiments (Waxm an and Smits, 1968). In particular,

both models are specific in their prediction of the effec-

tive transition from the non-linear to the linear zone

(Fig.

2),

an occurrence which was not presented as a

function of lithology. Extrapolation to other forma-

tions requires careful verification that the basic

assumptions of these m odels continue to be satisfied,

especially with regard to the concomitant representa-

tion of data from the non-linear and linear zones.

Another suite of double layer models, which has

received much less attention in the literature, is that

involving the surface area of pore systems. Rink and

Schopper (1974) proposed a m odel based on the specific

surface area of shaly reservoir rocks in which

(10)

where

Spo r

is the surface area per unit pore volume,

is the surface density of mobile charges, 6

is

the

effec-

tive mobility of these carrier charges within t he do uble

layer, and

X

is

a

tortuosity associated with the double

layer (Table

3).

Since the product a6 was proposed to be

approximately consta nt for a given cation, X was also

taken to be c onstant; therefore, the model was intended

to represent only the linear zone of Figure 2. Similar

comments can be applied to the related surface-con-

ductance model of Street (1961) an d to the surface-

structure m odel of Pape and Worthington (1983).

Discussion

T he field application of V , h models usually requires

that V , h be estimated a t each designated level using one

or more shale indicators.

A

shale indicator is simply a

conventional log

or log

combination whose response

equation(s) can incorporate a shale fraction term. Each

shale indicator is calibrated

so

that under ideal condi-

tions it furnishes a reasonable estimate of

vsh

Where

there are departures from the ideal conditions for a par-

ticular indicator, the resulting

v , h

is a n over-estimate. It

is the usual practice to obtain several estimates of v s h

from different shale indicators and then to select the

lowest value as the best estimate at

a

particular level.

This means that a log derived V s h might, for example,

have resulted from measurements

of

natural gamma

activity, thermal neutron population or sonic transit

time, quantities that bear little physical resemblance to

the resistivity-compatible parameter

X

of equation

(3).

Furthermore, as conditions change with depth, one

must expect the ideal shale indicator to c hange irregu-

larly. Thu s, not only might the derived v , h be physically

incompatible w ith t he parallel resistor equation

(3),

but

the degree of incompatibility can be expected to vary

erratically. Yet again, there is no guarantee that condi-

tions will be favourable at a given level for any of the

shale indicators used. It is, therefore, small wonder that

the V s h approach is widely regarded

as

deficient. Its sav-

ing grace has been that

V s h

is at least notionally log

derivable; an d for this reason, it has continued to retain

an important role in formation evaluation.

It has been argued that it is not

V s h

that should be

sought as a physical interpretation

of X ,

but rather an

effective shale volume fraction that takes account of

the composition, mode of distribution, and surface

geometry of c onstituen t shale. Thes e characteristics are

accommodated by models based on the ionic double

layer.

The double layer models do offer physical inter-

pretations

of

X that are electrically compatible, at least

in theory. Unfortunately, however, there are no estab-

lished techniques for the direct downhole m easurement

of

X as interpreted in these models, although

a

ray

of

promise in this direction lies in the recent application

of

frequencydom ain induced polarization to Q. determina-

tion (W axman and Vinegar, 1981).Nevertheless, because

these models represent X through electrochemical and

geometrical parameters that can be measured in the

laboratory, they would appear to afford a means of

calibrating a log-derivable petrophysical parameter in

terms of a n appropriate shale related quantity. Indeed,

the field application of the double layer models has

followed this very philosophy of indirectness. For

example, Lavers et a l. (1974) correlated

Qv

with porosity

for Nor th Sea reservoirs. Johnso n a nd Linke (1976)cor-

related cation exchange capacity with gamma ray

response. They used laboratory CEC data to derive a

method of determining effective shale volume from

gamma-ray response using a non-linear relationship. In

this way a double layer model was used to control t he

input to

a

v s h model. Yet again, Ju has z (1981) proposed

obtaining

Q.

from th e dry clay fraction, a parameter

which was determined using the neutron and density

log responses.

In

general, the need to correlate empiri-

cally

QY

or some related quantity with a log-derivable

parameter constitutes the major weakness of the

30

JANUARY-FEBRUARY,

1985


9/18

double layer models, which consequently have not had

the extensive impact within the industry that might

have been expected solely on scientific grounds.

While it is recognized that both the

vs

and the

double layer models suffer from deficiencies as regards

field application, both approaches are also seriously

affected by problems concerning laboratory measure-

ment. A determination of X in the laboratory can be

accomplished in two ways, by direct measurement of

the constituent parameters or by the multiple-salinity

indirect approach, whereby values of C, recorded at

several different values of C, are used to determine F,

and thence by calculation

X.

The direct laboratory

determination of vs is theoretically possible, but even

if it were meaningful there remains the problem of

C,. The indirect approach will provide a quanti-

tative estimate of some function of

vs

and C,,,

but it will not separately resolve these quantities. The

direct measurement of Spar and Q y is feasible, but it is

well known that different techniques furnish different

results (e.g., Van den Hul and Lyklema, 1968; Mian and

Hilchie, 1982). A decision is required as to which

measurement is likely to be the most meaningful in the

light of the intended application. Even

if

an appro-

priate measurement of Sporor

QV

an be made, the for-

mation in question might not satisfy the chosen model

and might therefore preclude a useful calculation of

X.

In this event, recourse can again be made to the

multiple-salinitymethod whereby X can be determined

as a composite term. Thus, for example, Kern et al.

(1976), recognizing that X-values based upon direct

measurements of

QY

were incorrect for certain tight gas

sands, concluded that these formations could not be

represented through the Waxman-Smits model and pro-

ceeded to determine

X

from an equation of the form of

(3).

The foregoing might appear as an unduly pessimistic

appraisal. Yet, it must be mentioned again that we have

up to now confined this treatment as far as possible to

cases of full water saturation. In the presence of hydro-

carbons, shale effects become more pronounced and

consequently the shaly-sand problem assumes even

greater degrees of significance and complexity.

APPLICATIONTOTHE HYDROCARBON ZO NE

The extension of the shaly-sand equations of Tables

2 and 3 to take account of the hydrocarbon zone

requires that terms in w be incorporated into the rela-

tionships. This has generally resulted in one of two dis-

tinct outcomes, a change in only the clean sand term

of the corresponding water-zone equation or a modifi-

cation of both the clean and shaly terms.

S, Equations for Shaly Sands

Changes to the clean term have been based upon

Archie's well known water saturation equation for

clean sands, viz.

(11)

c w "

c - -ss

where

C ,

is the conductivity of

a

reservoir rock that is

partially saturated to degree S , with electrolyte

of

con-

ductivity C,, and n is a clean sand saturation exponent

often taken to be two. Thus, if only the clean term is

changed, the general water-zone equation

(3)

can be

transformed to

' -

F

c,

=c-s, + x

(12)

so

that when X s very small or C, is very large, equa-

tion (12) reduces to the clean sand equation (11).

Specific examples of equation (12) are the Hossin (1960)

and Simandoux (1963) equations, Table 4, the latter

relating explicitly to values of S , above the irreducible

water saturation (Bardon and Pied, 1969).

TABLE 4

(HYDROCARBON ZONE)

SHALY-SAN D RELATIONSHIPS INVOLVING V,

c

=%s w %

csh

Hossin (1960)

F

F

c,

Ct

= +

xhcsh

Simandoux (1963)

c,

=

$ s

+Vsh csh s

Bardon

&

Pied (1969)

=

i?

-k

v h c Doll (unpublished)

< ;/* + y;-v+

s

Poupon and Leveaux

1971)

Changes to the shale term have usually had the

effect of introducing a factor S;where

s

can be loosely

regarded for our purposes as a shale-term saturation

exponent. Thus,

if

both the clean and the shale terms

are changed, the general water-zone equation (3) can

be transformed to

c,

Gs: + x

s:,

(13)

Again, this equation reduces to equation

(11)

when

X

is very small or C, is very large. Specific examples of

equation

(1

3) are the modified Simandoux equation

(Bardon and Pied, 1969) of Table 4 and the A. J de

Witte (1957), Waxman-Smits (1968) and Clavier et al.

(1977, 1984) equations of Table 5. Although not a

double layer model, the A. J de Witte equation has

been placed in Table 5 because of its correspondence to

the Waxman-Smits equation, as noted by Waxman and

Smits (1968) themselves.

THE LOG ANAL YST 31


10/18

T he remaining equatio ns of Table 4 can be reached

by initially taking the square root of each term in equa-

tions (12) an d (13), just

as

in t he water-zone case.

Where only the clean term is chan ged, we can modify

equatio n (12) to give

c ;I2

+ Jx (14)

Note that a very small X or a very large C, still causes

equation (14) to reduce to the clean sand equation

(11).

TABLE 5

SHALY-SAND RELATION SHIPS FOR

DOUBLE-LAYER MODELS

(HYDROCARBO N ZONE)

C,

= -S:

W + AS,

F

c, = W

s;

+ Q

s;-

F*

F*

A.

J. de Witte

1957)

Waxman

&

Smits

1968)

Clavier et al.

1977,

1984)

A specific example of equation (14) is the unpublished

Doll equation of Table 4 (cited by Desbrandes, 1968;

Raiga-Clemenceau, 1976).

Where both clean and shale terms are changed we

can modify equation (13) to give

(1

5 )

A

very small X or a very large C, causes equati on (15)

to reduce to equation

1

1).

A

specific example

of

equa-

tion (15) is the Indonesia fo rmula of Poupon and

Leveaux (1971) in Table 4.

J , /'

+

V'XS:12

Classification

of S,

Equations

Equ ations (12)-(15) epresent a four part family of

S,

equations for shaly sands. Most of the equations pro-

posed over the past 30 years can be identified with one

of

these four groups. Noteworthy exceptions are pro-

cedures which draw upon exponential functions (e.g.

Krygowski an d Pickett, 1978) an d certain eq uations of

strictly local application (e.g. Fertl and Hammack,

1971).

In using equations (12)-(15) as the basis for a

TABLE 6

TYPE EQUATIONS FOR S RELATIONSHIPS

TYPE EQUATION

NO. COMMENTS

1

c,

= ff S, + y

16) No

interactive term, S does not

appear in both terms

2. c = ff

s

+ y

s

3

c,

=

ff

S,

+

p

s

+

y

4

c,

= ff S, + p

s

+ y

s

17)

18)

19)

No interactive term, S appears

Interactive term,

S

does not

Interactive term, S appears in all

in both terms

appear in all terms

terms

a-denotes predominant sand term; P-denotes predominant interactive term; ?-denotes predominant shale term.

TABLE 7a

S, EQUATIONS

OF

TYPE

1

REFERENCE EQUATION COMMENTS

Laminated shale model;

F

=

formation factor of clean-sand

S,

relates to total interconnected pore

+

vshcsh

POUPON

l Xh) cw

s

et al

1954)

c,

=

F

streaks;

space

of

clean sand streaks

c w 2

c,

= w +

%

csh

F

OSSIN

(19601

SIMANDOUX

C, = %.

S +

EV,hCsh

1

963)

F

E

=

1 for high S,

E c 1 for low S,

F relates to free-fluid porosity unless otherwise stated; S relates to free-fluid pore space unless otherwise

stated; Equations are written with n

=

2.

32



11/18

classification scheme for S , equations, it is helpful to

rewrite equations (14)and

(15)

without the square root

function of C,. This not only facilitates comparisons

between the various expressions, but it also emphasizes

the presence within these equations of an interactive

sand-shale term encompassing S : , where r can be

regarded as an interactive-term saturation exponent.

On this basis we can use equations (12)-(15) s the foun-

dation for the four generalized type equations (16)-(19)

listed in Table 6.

The four type equations (16)-(19)describe categories

of relationships to which most of those S, expressions

reported in the literature can be assigned. Examples of

Types 1-4 are grouped in Tables 7a-7d, respectively.

Some of these expressions have been rearranged from

their conventional presentation in an effort to minimize

variations in format. Comments on certain points of

interest now follow.

The laminated shale model of Poupon et al. (1954) n

Table 7a might be classified as Type

3

when the com-

TABLE

7b

S EQUATIONS OF TYPE 2

REFERENCE EQUATION CO MMENTS

L. de WlTTE

2.15

k m s.

msh S,

m

=

molal concentration of

msh= molal concentration of

k = conversion from

m,.,,

to

F relates to total interconnected porosity

S relates to total interconnected pore

c,

= F F exchangeable cations in formation

1955)

water

exchangeable cations associated

with shale

conductivity

space

c, = w S, + A S,

. J. de WlTTE

1957) F

CLAVIER et al. c 2 C * W - C,) v Q" s

FO

1977, 984)

c =

w +

F O

cw 2

[

; -

]

vsh

4 s h

sw

4

=

+

c

UHASZ 1981)

F

See notes at foot of Table 7a.

THE LOG ANALYST

F = maximum formation factor

FA = shaliness factor

S


C, = conductivity due to shale ( # Csh)


S


F* relates to total interconnected

S,


Modified Simandoux equation

space

space

porosity

space

F relates to the free fluid porosity of

the total rock volume, inclusive of

intraformational (laminated) shales

Dual-water model


S, relates to total interconnected pore

Normalized Waxman-Smits equation

F =

l p

here

4

is the porosity derived

from the density log and corrected

for hydrocarbon effects

Fsh

=

1/42 here qjSh is the shale

porosity derived from the density log

S,


space

space

33


12/18

posite bracketed term is fully expanded into two com-

ponents. This has not been done primarily because in

this laminated model with clean sand streaks the quan-

tity I -

Vsh )

elates to the volume fraction of clean sand

within th e rock as a whole. It would be meaningless to

break this term dow n to yield an interactive term in

C,,

F

and

V,,

since the first two param eters relate to zones

which are stipulated to exclude altogether the shale

laminations.

The dual water model

of

Clavier et al. (1977, 1984)

in Table 7b m ight be classified as Type

4

with the com-

posite term in Cb,-C, ) fully expanded. There are two

reasons why this expression has been retained as Type

2.

Firstly, this brack eted term represents a concep tually

mea ningful excess water c onduc tivity. Secondly, expa n-

sion

of

the bracket would introduce a discrete negative

term which would be at variance with the concept of

resistors in p arallel.

A

similar line of rea soning can be

formulated for the Juhasz equation, also in Table 7b.

Table

7c


REFERENCE EQUATION COMM ENTS

Clay slurry model

F

relates to total volume occupied by

fluid and clay

S, relates to fluid-filled pore space

q(l- 9

csh

+

cw)

S, q2sh

w (1-q)2

s,

ALGER

et al (1963) c,

= F F

F

=

14; where

S

relates

to

total interconnected pore

is total interconnected

C

c, = ,

+

2

vsh

ANTON F

HUSTEN &

(1981)

c

=

c

+ v csh { -E[

space

Laminated sand-shale model

( l &h) cw

Si

ATCHETT

&

HERRICK

c,

=

F (

-

F B Q ~w + vsh csh v = volume fraction of laminated

(1982) shales only

F

relates to total interconnected porosity

within shaly-sand streaks

S,= relates to total interconnected

pore space within shaly-sand streaks

See notes at foot of Table 7a

TABLE 7d


R

EFE

R

ENC E

POUPON & C

V

, Indonesia formula

EQ UAT I0N

cr =

$ +

COMMENTS

LEVEAUX

F

(1971)

+ V i -

c s h si

WOODHOUSE c,=-s,

+ 2

(1976)

F

Modification of Poupon

&

Leveaux

equation for tar sands

34



13/18

The quantity B in the Waxman-Smits equation

(Table 7b) is a function of the bulk electrolyte conduc-

tivity

C,.

The product

BQY

s therefore intuitively an

interactive term. However, for classification purposes it

has been regarded strictly as a shale term since varia-

tions in

B

effectively determine the degree of manifesta-

tion of the cation exchange capacity and do not affect

the sand term directly.

The equation of Patchett and Herrick (1982) n Table

7c does not contain an interactive term in the sense of

the other equations in this group. It is in fact a com-

bination of the Waxman-Smits equation (Table 7b) and

the expression of Poupon et al. (1954) in Table 7a. The

effect of this combination is to produce an equation of

Type

3.

Discussion

We have introduced hydrocarbon zone equations by

considering initially relationships for water saturated

sands and then describing how these expressions have

been modified in order to arrive at the S, equations

that have been proposed in the technical literature.

As

implied earlier, this two stage breakdown was imposed

to facilitate the treatment and its understanding. How-

ever, some authors have proceeded directly to an

S,

equation, and in these cases, it has been necessary to

reduce their equations to water zone conditions retro-

actively, in order that the pattern of development might

be consistent within the overall scheme adopted here.

Thus, all

S,

equations are presumed to have been estab-

lished initially under water zone conditions and subse-

quently modified for use in the hydrocarbon zone.

In describing how these water zone equations have

been conceptually extended to take account of S, we

did not examine the reasoning behind each modifica-

tion. In certain cases the reasons have not been stated

explicitly, in others the approach has been solely empiri-

cal. Despite these shortcomings the following is an

attempt to piece together in skeletal form the thinking

behind the generalized family of equations (16)-(19),

using as a basis those cases where clear reasoning has

been presented.

Equations of Type 1 are usually based on

v s

models

(cf. Table 7a). The adoption of a fractional shale volume

and an intrinsic shale conductivity as the physical

interpretation of the shale parameter X does not make

any conceptual provision for the shale term to vary

with S,. This is because V,, is not an effective shale

volume (as per the modification of Johnson and Linke,

1976), but it is an absolute quantity. It is presumably

for this reason that the shale term is not a function of

S, in, for example, the equation of Hossin (1960). Yet

Simandoux did introduce some dependence on S, by

making provision for c s h to reduce through a coeffi-

cient

E ,

which falls below unity for saturations less than

some critical value of S,, corresponding to the amount

of water needed to saturate the double layer. This

reduction takes account of shale being isolated from

the conducting circuit as

S,

reaches very low values,

especially for reservoirs which are partially oil wet.

Equations of Type 2 are based on both

vh

models

and doublelayermodels (Table 7b). The shale term con-

tains

s,

and can therefore vary with water saturation.

In absolute terms, this variation is such that a reduc-

tion in

S,

leads to a reduction in the shale component.

In relative terms, however, a reduction in S, leads to an

increase in shale effects, since the sand component is

also reduced, but in proportion to S,. This projected

increase in shale effects with decreasing S, is not at

variance with Simandoux (1963), provided it is identi-

fied with values of S, greater than Simandouxs critical

water saturation. It is only as

S,

decreases below this

critical level that the Simandoux model predicts a de-

creasing shale effect.

In the case of Type-2

v s h

models Bardon and Pied

(1969), recognizing the difficulties of Simandouxs

approach, substituted

S,

for the coefficient

E

(Table 7a)

and thereby produced an equation

of

the form of

A.

J.

de Witte (1957). The object was to simplify the Siman-

doux equation

so

that greater ease of use compensated

for any resulting loss of accuracy. The development of

Type-2 equations does seem to have been guided by de

Wittes earlier work, especially since de Witte did not

restrict the approach by specifying a physical inter-

pretation of the general shale term A.

For Type-2 double-layer models, it has long been

recognized that a decrease in the amount of water

within the free fluid pore space causes an increase in

the relative importance of potential phenomena asso-

ciated with the double layer. This happens because

when free water is displaced by hydrocarbons, the

counterions must remain to ensure electro-neutrality.

The effect is to increase the shale term

X

o a new level

X

where

X

X =

,

(L.

de Witte, 1955; Hill

&

Milburn, 1956;

A . J

de

Witte, 1957; Waxman & Smits, 1968; Waxman &

Thomas, 1974). Thus, the enhancement of reservoir

rock conductivity, due to shale effects, can be expected

to be more pronounced for greater hydrocarbon satura-

tions. It is further reasoned that in the presence of

hydrocarbons, the geometric factor

F

must be replaced

by an analogous factor G which is related to F as

follows:

Thus, the general water-zone equation

(3),

rewritten as

c, = - I

(C,

+ FX)

F

THE LOG ANALYST 35


14/18

where th e bracketed term denotes th e equivalent water

condu ctivity C,, as in equation (9), must b e replaced by

(23)

Substituting for

X

from equation (20) and for from

equa tion (21) yields

1

Ct = (C, + FX)

c,

= -

w

s,

+

x

Y

(24)

which, with n

=

2, reduces to the form of the Type-2

double layer equations.

Equations of Type

3

(Table 7c) offer no consistent

reason why th e shale term does not contain S,. For the

equations of Doll and of Husten an d A nton (1981) the

absence

of S,

can be traced t o its absence in the square-

rooted equatio n (14). Since this equa tion was notionally

linked to equation (12) he reasons for the absence of S,

in the shale term are likely to be similar to those pro-

posed for the equations of Type-1. Th e form of the

equation of Alger et al. (1973) is a direct conse quence

of

the

clay

slurry model

(L.

de W itte, 1950) upon which

this expression is based. For the Patchett and Herrick

(1982) equation the shale term actually relates to

laminations, as per Poupon et al. (1954) whose model

made no provision for t he inclusion of S, in the shale

term (Table 7a).

The Type-4 equations involving v s h (Table 7d) all

have empirical origins. The dual porosity model of

Raiga-Clemenceau et

al.

(1984) is partly based upo n a n

empirical determination

of

the shale term saturation

exponent. This exponent turned out to be non-

trival, an outcome which has resulted in a Type-4

classification.

Despite the diverse origins

of

the equations

of

Table

7 the classification into type group s allows some order-

ing of what has hitherto been a highly disjointed sub-

ject area. As a consequence, apparently dissimilar

models can be seen to have common links from a

formation-evaluation standpoint. Th us, althou gh Table

7 contains only a proportion

of

those

S,

equations that

have been proposed over the years, there are grounds

for supposing that the interrelationship and corres-

pondence of the various

S,

equations might form a

basis for further developments leading towards an

improved conductivity model with a much wider and

more direct application.

F

FURTHER DEVEUlPMENaS

Before a prospective way forward can be meaning-

fully identified, even in broad terms, it is important to

be aware of th e general reasons for the multiplicity of

S, equations that is exemplified in Table 7. There are

two principal factors that have influenced the develop-

men t of these eq uations, (i) predictive performance

often in localized applications, an d (ii) the need for a

soun d scientific concep tual model. The developm ent of

any given S, equation has sometimes been dominated

by one of these factors at the expense of the other.

Predictive performance seem s to have been the main

prerequisite governing the emergence of v s h equations,

which have been progressively modified in an effort to

improve

local

accuracy. In this latter respect, a classic

development is the Indonesia formula of Pou pon and

Leveaux (1971). Because

V s h

equations lack a sound

scientific basis, and because they are necessarily founded

o n some localized control data, considerable disparities

can be expected between the estimates of S, that they

provide.

As

an example, Figure 5 contains comparisons

8

,\

60-

0

(P

4 0 -

E

B

v, 20-

v

0 20 40 6

80 100

S W H o s s i n [ I -

a1

0 2 0

4 0

60 80 100

s w D o l l

10/J

-

bl

Figure

5

Comparisons

of

predic ted water saturations

a) for the Simandoux and Hossin equation s,

and

b) for

the Simandoux and Doll equations

f rom

Fertl and Ham mack, 1971).

36 JANUARY-FEBRUARY,

1985


15/18

of the Simandoux and Hossin equations, both of

Type-1 (Table 7a), and of the Simandoux and Doll equa-

tions,

of

Type-l (Table 7a) and Type

3

(Table 7c), respec-

tively. It can be seen from Figure 5a that the Siman-

doux and Hossin equations show fair agreement at low

values of S,, i.e. around 20 saturation units. Yet, at high

values of

S,

there is a consistent disparity with the

Hossin equation furnishing values

of

S, that are some

20 saturation units greater than the Simandoux esti-

mates. In contrast Figure #5b ndicates that there is

generally good agreement between the Simandoux and

Doll equations for low values of

v s h .

However, for shale

fractions of 30 the Simandoux equation consistently

provides higher estimates of

S,

by some 15 saturation

units. All this illustrates that the disparity between

estimates is variable, and is strongly dependent on the

equations used and the prevailing values of v s h and s,.

These disadvantages can be partially compensated by

using C, as a tuning parameter to improve predictive

performance in the water zone in the expectation that

better estimates of S, will thereby be obtained in the

hydrocarbon zone.

The quest for a sound scientific conceptual model

has been responsible for the development of the double

layer equations. In this case certain aspects of predic-

tive performance have sometimes had to be sacrificed

in the interests of retaining a reasonable working

theory. Where inexactness has had to

be

admitted, it has

often been confined to petrophysical situations that are

less important from a formation evaluation standpoint

or are less likely to be encountered in practice, e.g. cases

of very high degrees of shaliness or exceedingly fresh

formation waters. An interesting example is to be found

in the reported mismatch of the dual water model to

experimental data for a very shaly sand, fully saturated,

with low salinity electrolyte (Clavier et al., 1984). In

this case the curvature associated with the calculated

trend of the dual water equation is actually opposite to

that of the experimental data trend, Figure 6. This is

partly a consequence of the dual water models inability

to track the data points corresponding to low values of

C,. The reason for this is that the model self-imposesa

lower limit of C, which corresponds to the case in

which the entire pore volume is occupied by bound

water of calculated conductivity 7.6 S m-l at 25C.

It is expected that the same influencing factors will

continue to govern the development of further

S,

equa-

tions, which will inevitably evolve especially as more

extreme environments are encountered. However, a

true solution to the shaly-sand problem will only be

achieved from an electrical standpoint when a sound

scientific theory gives rise to an S, equation which is

capable of a universally consistent predictive per-

formance. There is a further requirement that must also

be satisfied: the shale term(s) in an electrical S, equa-

tion must comprise log derivable parameters. These

requirements are not met by any of the

S,

equations in

routine use today.

The very nature of the type groups of Table 7 indi-

cates that many of the published S, equations are inter-

related and provides some basis for postulating the

existence of a shaly-sand algorithm with practically a

universal application. An encouraging indication of the

possible existence of an electrical relationship with

a

potentially wide application can be gleaned from

Figure 7. This diagram shows plots of

F,/F

vs C, for

four water-saturated sands of widely varying degrees of

shaliness. The significance of the ratio F,/F can be

appreciated from the following re-write of equation

(3):

It follows that F,/F is equal to the fraction of the total

conductivity that cannot be attributed to shale effects.

When this fraction is low, F,/F is low and shale effects

predominate. When this ratio is high, F, = F and shale

effects are not significant. For a shaly sand FJF can be

expected to vary with C,, decreasing as C, decreases

and free fluid conduction thereby becomes more

inhibited.

3o

- Q V = 1.47

meq c m - 3

F z 4 0 . 9

l

EXPANSION OF

DIFFUSE LAYER

I -

EXPERIMENTAL DATA

*

I

I

I

1

2 3 4

5

15

-1

C , I S m 1 -

Figure

6

Comparison of the dual-water model with

experimental data for a very shaly sand (from

Clavier et al.,

1984).

The four data plots of Figure 7 all show this trend

but are offset from one another within the range of

values of C,. Strong similarities are evident despite the

wide range of v represented, viz. 0.001-1.47 meq ~ m - ~

Furthermore, it can be envisaged that lateral displace-

ment of these curves could make them all virtually

coincident. This would appear to suggest that thesedata

distributions might all be described by a single algo-

rithm, provided that flexibility exists to account for

their different positions within the C, spectrum.

Moreover, the extension of these ideas to the hydro-

carbon zone follows directly (Worthington, 1982).

It is much more difficult to envisage how one might

satisfy the further requirement of log-derivable param-

THE

LOG ANALYST 37


16/18

t

Fa

F

1.0

0-95

0.1

0-01

~

Figure 7 Variation of F /F with C, for four diverse sandstone samples of very

different Q, (meq cm-9. Data from (1) Patnode and Wyllie (1950);

(2)

Wyllie and Southwick (1954);

3)

Rink and Schopper (1974); (4) Wax-

man and Smits (1968).

eters that characterize the electrical manifestation of

shaliness. T he c urrent lack of such

a

facility based o n

a

sound scientific theory constitutes one

of

the major

gaps in well logging technology. T he in dustry is pursu-

ing alternative strategies tha t might circumnavigate the

problem, e.g. induced gamma spectral logging and

dielectric logging, but neither of these has attained the

objective of furnishing a reliable, salinity independent

estimate of water saturation in shaly sands. As indi-

cated earlier, a most promising but physically difficult

approach to the fundamental problem

of

downhole

measurement

of

electrical shale parameters might be

found in induced polarization techniques, whose rele-

vance to formation evaluation continues to be empha-

sized (Worthington, 1984; Vinegar and Waxman,

1984).

CONCLUSIONS

In

1953 H .

G. Doll wrote, the m ost im portant prob-

lem th at has received thus far n o satisfactory solution

is that of shaly sands. Dolls comment is equally

applicable today. The shaly-sand problem

as

we know

it will not be solved until electrical shale parameters,

determined directly from downhole m easurements, can

be input to a reliable an d generally applicable predictive

algorithm

for

S,, that is based on a sound scientific

shaly-sand model. Fortunately, there are encouraging

signs that some progress is being made towards the

attainment of this recognized objective.

ACKNOWLEDGEMENTS

The author wishes to thank William R. Berry 11,

AndrC Hossin and Robert R. Kewley

for

helpful com -

ments during the preparation of the manuscript.

Publication

of

this work has been sanctioned by The

British Petroleum Company pic, whose support is

gratefully acknowledged.

NOMENCLATURE

shale term, A. J de Witte model (S m-l).

equivalent conductance of sodium clay-

exchange cations as a function of C,,

Waxm an-Smits model (S m2 eq-

x

10

-.

conductivity of bound water (S m-l).

conductivity

of

free or far water (S m-).

condu ctivity of fully water-sa ura ed rock

(S m-l).

conductivity due to shale, Patchett-Rausch

model (S m-l).

conductivity of wetted shale (S m-l).

conductivity of partially water-saturated

rock

(S

m-l).

condu ctivity of (free) water

S

m-l).

equivalent conductivity of total pore water

(S m-l).

cation exchange capac ity (meq l lOOg rock).

formation factor.

38



17/18

apparent formation factor.

formation factor, dual-water model.

formation factor of shale.

formation factor, Waxman-Smits model.

geometric factor for the electrical measure-

ment of effectively clean partially water-

saturated rock.

cation exchange capacity per unit pore

volume (meq ~ m - ~ ) .

resistivity of fully water-saturated rock@

m).

resistivity of shale

0

m).

resistivity of partially water-saturated rock

Q

4.

resistivity of (free) water

(Q

m).

pore surface area per unit pore volume (m-l).

fractional water saturation of pore space.

fractional wetted shale volume of rock.

generalized electrical parameter of shaliness

S m-l).

generalized electrical parameter of shaliness

in partially water-saturated rock S m-l).

coefficient in generalized Archie

F

-

4

equation.

shale factor, Hill-Milburn model.

conversion from mshand m, to conductivity,

L.

de Witte model S m-I mole l kg solvent).

exponent in Archie F -

4

equation.

molal concentration of exchangeable ca-

tions associated with shale,

L.

de Witte

model (moles/kg solvent).

molal concentration of exchangeable cations

in formation water,

L.

de Witte model

(moles/kg solvent).

saturation exponent, sand term.

shale volume fraction of pore space, clay

slurry model.

saturation exponent, interactive term.

saturation exponent, shale term.

amount of clay water associated with one

milliequivalent of clay counterions (meq-

cm3).

excess double-layer conductivity, Winsauer-

McCardell model S m-I).

predominant clean-sand term, type equa-

tions

S

m-l).

predominant sand-shale interactive term,

type equations S m-l).

predominant shale term, type equations

S m-l).

charge mobility in double layer (m2 V- s-

1.

shale-term coefficient, Simandoux model.

tortuosity associated with the double layer,

Rink-Schopper model.

surface charge density of rock C m-2).

fractional porosity.

bound-water porosity.

4e

effective (free-fluid) porosity.

4 s h

4t

total interconnected porosity.

total porosity of shale, Juhasz model.

REFERENCES

ALGER, R. P.,RAYMER, L. L., HOYLE, W. R., and TIXIER,

M.

P.

1963. Formation density log applications in liquid-

filled holes. J. Pet. Tech. 15, 321-332.

ARCHIE, G. E. 1942. The electrical resistivity log as an aid

in determining some reservoir characteristics. Trans.

BARDON, C. and PIED, B. 1969. Formation water saturation

in shaly sands. Trans. SPWLA 10th Ann. Logging Symp.,

21-19.

CLAVIER, C., COATES, G. and DUMANOIR,

J.

1977. The

theoretical and experimental bases for the dual water

model for the interpretation of shaly sands. SPE Paper No.

6859.

CLAVIER, C., COATES, G. and DUMANOIR, J. 1984.

Theoretical and experimental bases for the dual-water

model for interpretationof shaly sands. SOC.Pet. Engrs. J.

DESBRANDES,

R.

1968. Theorie et interpretation des

diagraphies. Editions Technip, Paris, 545 pp.

DE WITTE, A.

J.

1957. Saturation and porosity from electric

logs in shaly sands. Oil & Gas J.

55,

4 March 89-93, 15

April 115-121.

DE WITTE, L. 1950. Relations between resistivities and fluid

contents of porous rocks. Oil

&

Gas J. 49, 24 August

DE WITTE, L. 1955. A study of electric log interpretation

methods in shaly formations. Trans. AlME 204, 103-110.

FERTL, W. H. and HAMMACK, G. W. 1971. A comparative

look at water saturation computations in shaly pay sands.

Trans. SPWLA. 12th Ann. Logging Symp., R1-17.

HILL, H. J. and MILBURN, J. D. 1956. Effect of clay and

water salinity on electrochemical behavior of reservoir

rocks. Trans. AlME 207, 65-72.

HOSSIN, A. 1960. Calcul des saturations en eau par la

methode du ciment argileux (formule d Archie

generalisee). Bull. Assoc. Francaise Tech. Pet. 140, 31

March.

HUSTEN, P.and ANTON, H. 1981. The porosity surface con-

cept. Oil Gas 7(1), 40-42.

JOHNSON, H. M. 1961. A history of well logging. Trans.

SPWLA 2nd Ann. Logging Symp., Paper 1.

JOHNSON, W. L. and LINKE, W.A. 1977. Some practical

applications to improve formation evaluation of sand-

stones in the Mackenzie Delta. Trans. CWLS 6th Formation

Evaluation Symp., R1-19.

JUHASZ, 1 1981. Normalised

Q,

he key to shaly sand

evaluation using the Waxman-Smits equation in the

absence of core data. Trans. SPWLA 22nd Ann. Logging

Symp., 21-36.

KERN,

J.

W., HOYER, W. A. and SPANN, M. M. 1976. Low

porosity gas sand analysis using cation exchange and

dielectric constant data. Trans. SPWLA 17th Ann. Logging

Symp. PP1-17.

KRYGOWSKI, D.A. and PICKETT, G.

R.

1978. The prediction

of water saturation and recoverable hydrocarbon volumes

in shaly sand reservoirs. Trans. SPWLA 18th Ann. Logging

Symp., 881-15.

LAVERS, B. A., SMITS, L. J. M. and VAN BAAREN, C. 1974.

Some fundamental problems of formation evaluation in

the North Sea. Trans. 3rd European Formation Evaluation

Symp., SPWLA London Chapter, G1-15.

AlME 146, 54-62.

24, 153-167.

120-132.

THE LOG ANALYST

39


18/18

MIAN, M. A. and HILCHIE, D. W., 1982. Comparison of

results from three cation exchange capacity CEC)

analysis techniques. The

Log

Analyst 23 5), 10-16.

PAPE, H. and WORTHINGTON,

I?

F. 1983. A surface-

structure model for the electrical conductivity of reservoir

rocks. Trans. 8th European Formation Evaluation Symp.,

SPWLA London Chapter, 21-19.

PATCHETT, J. G. and HERRICK, D. C. 1982. A review of

saturation models. In: Shaly Sand Reprint Volume,

SPWLA, Houston, pp 111 1-7.

PATCHETT,J. G. and RAUSCH,

R.

W 1967. An approach to

determining water saturation in shaly sands. J. Pet. Tech.

PATNODE, H. W and WYLLIE, M. R.

J.

1950. The presence

of conductive solids in reservoir rock as a factor in electric

log interpretation. Trans. AlME 189, 47-52.

POUPON, A. and LEVEAUX, J. 1971. Evaluation of water

saturations in shaly formations. Trans. SPWLA 12th Ann.

Logging Symp., 01-2. Full text in Shaly Sand Reprint

Volume, SPWLA, Houston, pp IV 81-95).

POUPON, A., LOY, M. E. and TIXIER, M.

F?

1954. A con-

tribution to electric log interpretation n shaly sands. Trans.

RAIGA-CLEMENCEAU, J. 1976. A quicker approach of the

water saturation in shaly sands. Trans. SPWLA 17th Ann.

Logging Symp., El-16.

RAIGA-CLEMENCEAU, J., FRAISSE, C. and GROSJEAN, Y.

1984. The dual porosity model, a newly developed inter-

pretation method for shaly sands. Trans. SPWLA 25th

Ann. Logging Symp., F1-16.

RINK, M. and SCHOPPER,

J.

R. 1974. Interface conductivity

and its implications to electric logging. Trans. SPWLA 15th

Ann. Logging Symp., J1-15.

SAUER, M. C., SOUTHWICK,

F

F., SPIEGLER, K. S. and

WYLLIE, M. R. J. 1955. Electrical conductance of porous

plugs: ion exchange resin-solution systems. Ind. Engng.

Chem. 47, 2187-2193.

SCHLUMBERGER. 1972. Log Interpretation, Volume 1

-

Principles. Schlumberger Ltd., New York, 113 pp.

SIMANDOUX, P 1963. Dielectric measurements on porous

media: application to the measurement of water satura-

tions: study of the behaviour of argillaceous formations.

Revue de I Institut Francais du Petrole 18, supplementary

issue, 193-215. Translated text in Shaly Sand Reprint

Volume, SPWLA, Houston, pp IV 97-124).

STREET, N. 1961. Surface conductance and the conductive

solids effect. Illinois State Geol. Surv. Circ. 315, 16 pp.

VAN DEN HUL, H. J. and LYKLEMA, J. 1968. Determination

of specific surface areas of dispersed materials: com-

parison of the negative adsorption method with some

other methods. J. Am. Chem. SOC.90, 3010-3015.

VINEGAR, H.

J.

and WAXMAN, M. H. 1984. Induced polari-

zation of shaly sands. Geophysics 49, 1267-1287.

WAXMAN

Documents

Worthington,P.F.,1985 -The Evolution of the Shaly-Sand Concepts in Reservoir Evaluation