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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
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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
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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
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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.
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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
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.
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
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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
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..
.
.
.
.
.
~
.
.~
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
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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.
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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.
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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
relates to total interconnected pore
C, = conductivity due to shale ( # Csh)
F relates to total interconnected porosity
S
relates to total interconnected pore
F* relates to total interconnected
S,
relates to total interconnected pore
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
F relates to total interconnected porosity
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,
relates to total interconnected pore
space
space
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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
S EQUATIONS OF TYPE 3
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
S EQUATIONS OF TYPE 4
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
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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
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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,
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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-
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LOG ANALYST 37
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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
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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.
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