Method apparatus for determination of porosity lithological composition

Evaluation of thinly laminated shaly sand reservoirs has long been one of the most difficult problems of log analysis. A primary reason is that only shallow shale indicators such as a Dipmeter, other microresistivity devices, or an ultra high frequency dielectric tool, etc. accomplish resolutions compatible with the most thinly bedded shale or sand laminae. To overcome this problem a technique has been developed to reconstruct deep Induction conductivity and to compute effective porosity and water saturation consistent with the high vertical resolution tools such as the Dipmeter. To achieve greater accuracy in the evaluation of shale content and porosity, the volumes of shale are initially estimated from both a density-neutron crossplot and a high resolution shale indicator which has been integrated to the vertical resolution of the density and neutron logs. Then shale parameters for these logs are automatically adjusted within limits suggested by log data in such a way that computed shale volumes from the shale indicator and density-neutron crossplot match each other. The adjusted parameters are used to compute porosity and shale volume and the mode of distribution from the density and neutron logs and to recompute these results to the high vertical resolution level. This information is in turn used to reconstruct the deep Induction conductivity to the same vertical resolution. The technique allows water saturation determination from a Waxman-Smits type model when both dispersed and laminated clay types are present.

BACKGROUND OF THE DISCLOSURE 
Most logging tools measuring porosity, resistivity, radioactivity and so on 
can record only average values of sand and shale properties in thinly 
laminated reservoirs. Only logging tools with very small radii of 
investigation such as the dipmeter or high frequency dielectric tool have 
vertical resolutions compatible with thicknesses of separate beds. These 
shallow investigation devices can be used to delineate thin beds within 
laminated reservoirs, and to determine shale volume. Such data are 
relatively shallow. However, evaluation of properties such as porosity, 
resistivity, and water saturation of these thin beds at greater distances 
has to be accomplished using logs with degraded vertical resolution, in 
combination with the high resolution shale indicator. This is possible 
mainly because the laminated shaly sand reservoir is a two component 
system (strata or laminar layers of sand and shale), and a change in 
properties of each of these components within the vertical resolution of 
most tools is rather insignificant compared to the drastic differences in 
properties between two components (shale versus sand) and therefore 
changes in the composite properties of the laminated reservoir. Since a 
sand component property and the logs with degraded vertical resolution 
usually have similar frequency characteristics, the former can be 
adequately restored from the logs. Subsequently, the high frequency 
composite log or formation property can be reconstructed from the 
component data and a high frequency shale indicator. 
Several different methods of computing high resolution deep resistivity and 
other laminated reservoir parameter appear in the literature, Laminated 
Sand Analysis, D. F. Allen, SPWLA 25 Logging Symposium, Jun. 10-13, 1984; 
Comparative Results of Quantitative Laminated Sand Shale Analysis in Gulf 
Coast Wells Using Maximum Diplog Microresistivity Information, T. H. Quinn 
and A. K. Sinha, SPWLA 26th Annual Logging Symposium, Jun. 17-20, 1985; 
and Taking Into Account The Conductivity Contribution of Shale Laminations 
When Evaluating Closely Interlaminated Sand-Shale Hydrocarbon Bearing 
Reservoirs, J. Raiga-Clemenceau, SPWLA 29th Annual Logging Symposium, Jun. 
5-8, 1988. The present technique has some features in common with these 
prior publications, but the overall teaching hereof is unique. 
In laminated reservoirs, even productive ones, the volume of shale can 
exceed the volume of sand, and thus the volume of shale computation can 
strongly impact all subsequent evaluations. Accurate determination of 
shale volume is therefore significant. This approach matches shale volumes 
computed from one of several integrated high resolution shale indicators 
and a density-neutron crossplot by automatically adjusting shale 
parameters. These parameters are used to compute volumes of shale at two 
levels of vertical resolution; one of density and neutron logs and a 
second being a high resolution shale indicator. All other properties are 
also computed at two levels of vertical resolution, the first at low 
vertical resolution to compute components from composite properties and, 
the second at the high resolution to compute composite properties from 
sand and shale components. If the computations at lower resolution require 
different vertical resolutions, the logs with the higher vertical 
resolution are integrated to lower vertical resolution. Most existing 
water saturation equations, including the Waxman-Smits model, evaluate 
reservoirs having only dispersed clay. In the technique described below, 
the Waxman-Smits model is used to compute water saturation in the 
laminated shaly sand reservoirs but its parameters are adapted and used 
accordingly. 
The disclosed technique works best in areas where properties of the sand 
and shale components of the laminated reservoirs have low frequency 
character, i.e. where variations within the sand and shale components 
occur much more slowly than variations in the composite character of the 
beds shown by high resolution logs (and caused by changes in the 
distribution of the laminae). This pattern is common for reservoirs in 
which effective porosity depends mainly on volume and mode of clay 
distribution. The Gulf Coast and similar areas where such laminated 
productive reservoirs are widely developed are primary targets for the 
technique described below.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
Before beginning with the description of the present disclosure, it is 
helpful to set out the following nomenclature. These are symbols used 
hereinafter: 
B equivalent conductance of clay exchange cations, liter/eqv ohm m 
B.sub.c compound equivalent conductance of clay exchange cations, liter/eqv 
ohm m 
B.sub.sd B in sand laminae, liter/eqv ohm m 
B.sub.sh B in shale laminae, liter/eqv ohm m 
CEC cation exchange capacity, meq/100 g rock 
C.sub.o salinity of equilibrating NaCI solution, eqv/liter 
C.sub.sh conductivity of shale component,mho/m 
C.sub.sd conductivity of sand component, mho/m 
C.sub.t measured formation conductivity, mho/m 
C.sub.w water conductivity in sand laminae, mho/m 
C.sub.wc compound water conductivity, mho/m 
C.sub.wsh water conductivity in shale laminae, mho/m 
D.sub.cl clay density, gm/cc 
Den density log 
D.sub.sh shale density, gm/cc 
D.sub.g sand matrix density, gm/cc 
F* formation resistivity factor in Waxman-Smits model 
m cementation exponent 
n saturation exponent 
N.sub.cl neutron log response to clay 
N.sub.eu neutron log 
N.sub.sh neutron log response to shale 
Q.sub.v concentration of clay exchange cations per unit pore volume, meq/ml 
Q.sub.vd Q.sub.v related to dispersed clay, meq/ml 
Q.sub.vl Q.sub.v related to laminated clay, meq/ml 
R.sub.t measured formation resistivity, ohmm 
R.sub.w water resistivity in sand laminae, ohmm 
R.sub.wsh water resistivity in shale laminae, ohmm 
S.sub.w effective water saturation 
S.sub.wt total water saturation 
S.sub.wts total water saturation in sand laminae 
V.sub.cl volume of clay 
V.sub.cld volume of dispersed clay 
V.sub.cll volume of laminated clay 
V.sub.sh volume of shale 
V.sub.shl volume of laminated shale 
.phi..sub.cld fraction of total porosity related to V.sub.cld 
.phi..sub.e effective porosity 
.phi.max porosity in clean sand 
.phi..sub.sh shale porosity 
.phi..sub.shl fraction of total porosity related to V.sub.shl 
.phi..sub.t total porosity 
.phi..sub.ts sand component of total porosity 
.phi..sub.vcl fraction of total porosity related to V.sub.cl 
.phi..sub.vsh fraction of total porosity related to V.sub.sh 
Additional subscripts are worth noting, namely a high resolution value is 
indicated by the subscript hr, while reconstructed value is simply 
indicated by the subscript r. 
THE LAMINATED RESERVOIR AS A TWO COMPONENT SYSTEM 
From the log analysis point of view, the thinly laminated reservoir is a 
reservoir where alternating sand and shale beds are quite thin, 
considerably thinner than the vertical resolutions of most logging tools 
including those which provide data for computing porosity, water 
saturation and other reservoir parameters. Several basic tools and their 
vertical resolution levels are presented in Table 1 below. All tools can 
be grouped into three basic groups. Each group has a certain or specific 
vertical resolution which are high (dipmeter, microlog, high frequency 
dielectric, etc.), medium (density, neutron, acoustic) and low resolution 
(induction, laterlog). High vertical resolution is from one to several 
inches and generally has very small radii of investigation. In this 
disclosure, the high resolution tools need only indicate shale. If the 
resolution is only two or three feet vertical resolution, the data is 
sufficient to enable volume of shale, the shale distribution, porosity, 
cation exchange capacity and Q.sub.v to be computed. A low vertical 
resolution distinguishes a few feet for the recently introduced High 
Resolution Induction Tool (Halliburton Logging Services, Inc.) to eight 
feet or more for most older induction logging tools. 
TABLE 1 
______________________________________ 
VERTICAL RESOLUTION LEVELS OF LOGGING TOOLS 
Resolution Level 
Property Obtained 
Logging Tool 
______________________________________ 
high shale volume dipmeter, microlog, 
high frequency 
dielectric, 
unfiltered Pe-index 
medium shale volume, shale 
porosity logs including 
distribution, porosity 
density, neutron, 
Q.sub.v and CEC 
acoustic 
low conductivity, induction, laterlog 
resistivity 
______________________________________ 
By definition, only the high resolution tools can detect the separate 
(sand/shale) beds in the thinly laminated reservoir. All other tools form 
average or composite signals from both sand and shale laminae together, 
mainly as a result of the broad vertical resolutions of the tools. The 
resolution can vary over a range of about one hundred fold or more. To 
achieve accurate log interpretation, signals from the sand and shale beds 
have to be separated. This separation can be done because component 
properties of the individual sand and shale beds are nearly uniform and 
usually change only gradually. Thus, if ten sand layers are considered, 
they are usually similar in most characteristics. This is not true of the 
composite values from ten sand layers and ten shale layers, measured by 
these tools, which measured values can change drastically when crossing 
boundaries between sand and shale laminae. Thus the individual layer 
component properties are preserved better than composite properties in the 
average values measured by logging tools with medium or low vertical 
resolution. 
Directing attention to FIG. 1a of the drawings, a laminated shaly sand 
reservoir is shown to have equal thicknesses of sand and shale beds. FIG. 
1b shows a measurable property X of this reservoir. The property X is any 
measurable property typically measured by a logging tool. The property or 
measurable parameter has low and constant value in shale beds and high and 
somewhat uneven values in the sand beds. A continuous curve at FIG. 1c 
through the sand values represents this property, while a straight line 
connecting shale values is the shale component. The curve shape is 
dependent on tool resolution. Assume that a tool used to measure this 
property has a low vertical resolution, which is several times wider than 
the thicknesses of the beds, and that within the vertical resolution of 
the tool all beds contribute to the measured signal in direct proportion 
to their component values regardless of their distances from the measure 
point. Also assume that all beds beyond tool vertical resolution do not 
contribute to the signal at all. Then a composite curve X.sub.log 
"measured" by this tool can be computed as follows: 
EQU X.sub.log =X.sub.sd .times.(1-V.sub.shl)+X.sub.shl .times.V.sub.shl(1) 
X.sub.sd, X.sub.shl, and V.sub.shl are the average value of the sand 
component of property X, the average value of laminated shale component of 
property X, and the average laminated shale volume respectively, within 
vertical resolution of the logging tool. 
At the laminated formation 10, the width of tool response is crisply 
defined by three brackets 11, 12 and 13 which represent differing tool 
resolutions, the bracket 11 being a typical high resolution range of 
investigation. Indeed, the bracket 13 can be one hundred times wider than 
the bracket 11. The curve 14 shows the property X as measured by the tool 
sensitive to the property X as it is brought near the laminated formation 
10. The curve 10 is partially dependent on the investigative tool 
resolution. 
As further shown in FIG. 1c, the curve 16 is the continuous curve which is 
drawn through the sand values from the curve 14 therebelow. By contrast, 
the curve 17 is the shale curve of the property X and is a straight line 
because the shale value of X is low and fairly uniform. 
The sand component X.sub.sdr of property X is the curve 18 which is 
reconstructed from equation 2: 
##EQU1## 
The "measured" X.sub.log curve 20 is only half the initial sand component 
X.sub.sd, while the reconstructed sand component X.sub.sdr coincides with 
X.sub.sd on the slopes and is about 15% less at maximum of X.sub.sd curve. 
As could be expected, modeling with other examples of properties has shown 
generally that smoother and wider curves of the initial property value X 
can be more accurately reconstructed. Since laminated reservoir usually 
consist of many thin sand and shale laminae where the sand and shale are 
reasonably consistent, and, since the properties of adjacent sand laminae 
do not change significantly, the sand component can be reconstructed in 
most cases quite satisfactorily. 
The volume of laminated shale V.sub.shl measured by a high resolution shale 
tool is shown at FIG. 1d where the curve 22 is used along with the sand 
property X.sub.sdr to reconstruct the composite property shown in FIG. 1e. 
The high vertical resolution curve 23 of X.sub.rhr is obtained as follows: 
EQU X.sub.rhr =X.sub.sdr .times.(1-V.sub.shlhr)+X.sub.sh .times.V.sub.shlhr(3) 
SHALE VOLUME MEASUREMENTS 
Shale volume and distribution are important to this technique. They are 
computed at all three levels of tool vertical resolution. At the low level 
the laminated shale volume is used to compute sand component resistivity. 
At the medium resolution level, the volume of shale is used to determine 
formation porosity and the mode of shale distribution, namely, whether or 
not it is dispersed or laminated. At the high resolution level, shale 
volume and distribution are needed to recompute porosity, reconstruct high 
resolution composite resistivity and to determine water saturation. 
Shale volume computed from a high resolution shale indicator can be 
integrated to the medium resolution level, from which the volume of 
laminated shale can be integrated to the low level. This procedure 
(converting all data for different tool resolutions) can result in loss of 
accuracy because the integrated log response may not be exactly equal to a 
linear combination (a simple summation) of the values of the individual 
components for several reasons. First, the vertical resolutions of logging 
tools are known only approximately and therefore integration of shale 
volume from one level to another can produce inaccurate shale volume at 
the lower resolution levels. Second, determination of shale parameters is 
often subjective and to some degree depends on the experience of the log 
analyst. Third, no shale indicator is perfect. Shale parameters can change 
due to various geological factors such as presence of hydrocarbons, or 
variations in shale density, porosity and hence resistivity. That is one 
reason it is desirable to check at least two shale indicators against each 
other and adjust shale parameters if necessary. 
To increase the accuracy of the computed shale volume, a technique has been 
developed to automatically adjust shale parameters by comparing the high 
resolution shale volume integrated to the medium resolution level and the 
shale volume computed from density-neutron crossplot. The basic feature of 
this technique is solving two shale volume equations. This requires the 
density-neutron crossplot solution of FIG. 2 to be in the form used to 
compute shale volume from a conventional shale indicator. Thus, a neutron 
response to clean sand N.sub.sd is found at the intersection 24 of the 
clean sand line and a line 25 passing through the log value and wet shale 
point. Then the first equation can be written as: 
##EQU2## 
Where R.sub.xi is the integrated high resolution log used as a shale 
indicator, R.sub.sh is the high resolution log response in shale; and 
R.sub.sd is the high resolution log response in sand. Another equation is 
an equation of proportionality. Parameters can be changed within certain 
limits defined by their maximum and minimum values. The shale parameter, 
having greater possible deviation, will change more and one can therefore 
define equation 5: 
##EQU3## 
Solving the two equations 4 and 5 together, the two unknowns R.sub.sh and 
N.sub.sh can be found. At this stage, the parameter N.sub.sh is located by 
a point on the line 26 connecting log and shale points. This point is 
projected parallel to the clean sand line 17 onto the line 26 connecting 
the labelled maximum and minimum values of density and neutron shale 
parameters and the intersection 28 finally defines both parameters 
N.sub.sh and D.sub.sh in FIG. 2. Adjusted shale parameters from the point 
28 and corrected parameter R.sub.sh are then used to compute volumes of 
shale on both high and medium resolution levels. 
Since the presence of light hydrocarbons (e.g., CH.sub.4) can influence 
density and neutron log responses, for an accurate shale volume 
determination, logs have to be corrected if the density of light 
hydrocarbons in the formation is known. A high resolution shale indicator 
such as a dipmeter has a very small radius of investigation of one to two 
inches; in highly porous and permeable formations, the dipmeter measures 
only the flushed zone. Thus the influence of hydrocarbons on such a log is 
often negligible. If density and neutron logs cannot be corrected for 
light hydrocarbons, the volume of shale in hydrocarbon bearing zones 
should be determined only from a high resolution shale indicator (e.g., 
dipmeter) using a shale parameter which can be adjusted in a water bearing 
interval. 
Sand parameters cannot be corrected simultaneously during the adjusting of 
shale parameters. However, if the required adjustment of shale parameters 
exceeds minimum or maximum limits (see FIG. 2), then the next step is to 
adjust the sand parameters. This usually occurs in zones of low shale 
content where incorrectly defined sand parameters could result in very 
large corrections in the shale parameters. In water and oil bearing 
reservoirs, only the sand parameter of the high resolution shale indicator 
should be adjusted because density-neutron crossplot definitively locates 
the clean sand line except in cases where the density of sand matrix 
or/and water changes. Then, the sand parameter of the high resolution 
shale indicator is used to determine the volume of shale in zones having 
light hydrocarbons. The technique of adjusting sand parameters is similar 
to the one described for adjusting shale parameters. 
FORMATION POROSITY AND MODE OF CLAY DISTRIBUTION 
Medium Vertical Resolution Level 
After shale volume is determined (best done by the foregoing approach), the 
effective and total porosities at the medium level can be computed 
conventionally from the density-neutron crossplot. Then shale porosity 
(bound water content) is defined by equation 6: 
##EQU4## 
One method of determining the clay distribution with all three types of 
clay (dispersed, laminated and structural) present was developed earlier 
by Ruhovets, N. and Fertl W. H., Digital Shaly Sand Analysis Based on 
Waxman-Smits Model and Log-Derived Clay Typing, 7th European Logging 
Symposium, Oct. 21-12, 1981, Paris France. Since structural clay occurs 
rarely, only the dispersed and laminated clays are significant to this 
disclosure. For most geological conditions, derivation of an equation for 
the clay distribution is given by equations 7 through 10: 
EQU .phi.e=.phi.max (1-V.sub.shl)-V.sub.shd (7) 
If only laminated and dispersed clays are present, one obtains: 
EQU V.sub.cld =V.sub.sh -V.sub.shl (8) 
EQU .phi..sub.e =.phi.max (1-V.sub.shl)-V.sub.sh +V.sub.shl (9) 
##EQU5## 
V.sub.cld is determined from equation 8. 
Laminated shale is a detrital (allogenic) deposit, and thus always contains 
not only clay minerals but also other fine grained materials such as silt, 
carbonates, organic matter and so on. According to Yaalon, D. H., Clay 
Minerals Bull., 5(27), 31-6, 1962, based on analysis of 10,000 shales, 
clay minerals constitute just under 60% of average shale. Thus laminated 
clay is only a part (e.g., almost 60%) of laminated shale. On the 
contrary, dispersed clay is primarily authigenic clay minerals formed 
after the sand was deposited, Visser, R., Bours, K. A. T., van Baaren, J. 
P., Effective Porosity Estimation in the Presence of Dispersed Clay, SPWLA 
29th Logging Symposium, Jun. 5-8, 1988. The percentage of clay in shale 
(or clay/shale ratio) is an important regional parameter (CLSH) in the 
present technique. If this parameter has not been determined for the 
selected area of investigation, a default value of 60% is usually assumed. 
Shale parameters such as D.sub.sh and N.sub.sh are usually determined in 
intervals of 100% shale. These parameters depend on clay content in shale 
and properties of clay and non-clay (mainly silt) materials. If the 
average density and hydrogen index of dry clay are known,then the 
parameters of wet clay, such as D.sub.cl and N.sub.cl, are found by 
extending the straight line 30 connecting the sand matrix point and the 
wet shale point on the density-neutron crossplot in FIG. 3. Then the 
clay/shale ratio (CLSH) can be determined in the following manner: 
##EQU6## 
Correspondingly, if the clay/shale ratio is known, the parameters N.sub.cl 
and D.sub.cl can be determined from equations 11 and 12. 
Since parameters D.sub.sh and N.sub.sh are determined in 100% shale where 
dispersed clay is not present, use of these parameters to compute volume 
and mode of clay distribution in shaly sand reservoirs can be erroneous. 
Actually, if only laminated shale is present in a shaly reservoir, 
D.sub.sh and N.sub.sh are used to determine the volume of shale. If only 
dispersed clay is present, D.sub.cl and N.sub.cl are used to determine the 
volume of clay. If both dispersed clay and laminated shale are present in 
the shaly reservoir, density and neutron parameters should be somewhere 
between these two sets of shale and clay parameters. Thus, after the 
volume of shale and the mode of clay distribution are determined, the 
parameters D.sub.sh and N.sub.sh are corrected through an iteration 
process. 
The iteration process includes the following six steps after shale volume, 
effective porosity, and mode of clay distribution are computed using 
original parameters D.sub.sh and N.sub.sh : 
Step 1. Volume of clay (V.sub.cl) is found: 
EQU V.sub.cl =V.sub.cld +CLSH.times.V.sub.shl (13) 
Step 2. Clay/shale ratio of the reservoir (CSR) is determined: 
##EQU7## 
Step 3. Parameter N.sub.sh is corrected: 
EQU N.sub.shc =CSR.times.N.sub.cl (15) 
Step 4. Correct the parameter D.sub.sh similarly: 
EQU D.sub.shc =D.sub.g -CSR.times.(D.sub.g -D.sub.cl ) (16) 
Step 5. The new volume of shale is determined from the density-neutron 
crossplot using corrected parameters N.sub.shc and D.sub.shc (See FIG. 3). 
Step 6. New V.sub.shl and V.sub.cld are computed, ending one iterative 
loop. 
The iteration continues (steps one to six) until V.sub.sh or one of the 
other variables such as CSR or N.sub.sh converges. Usually about five 
iterations are required. 
High Vertical Resolution Level 
Computation of total and effective porosities and the clay distribution 
mode at the high resolution level is performed through computation of the 
sand component of total porosity. The easiest way is to compute the high 
resolution effective porosity from the total porosity as follows: 
EQU .phi..sub.ehr =.phi..sub.t -.phi..sub.sh .times.V.sub.shhr (17) 
However, in this case one must assume that not only the total porosity does 
not change significantly within the vertical resolution of the porosity 
log, but also one must assume that the total porosities of the sand and 
shale laminae are approximately equal; the latter is often a poor 
assumption. It is much wiser to assume that the sand component of total 
porosity does not change significantly within the vertical resolution of 
the porosity tool. This component can be found from equation 18: 
##EQU8## 
If the volume of laminated shale from high resolution log is known, the 
total and effective porosities are found from equations 19 and 20: 
EQU .phi..sub.thr =.phi..sub.ts .times.(1-V.sub.shlhr)+(.phi..sub.sh 
.times.V.sub.shlhr) (19) 
EQU .phi..sub.ehr =.phi..sub.thr -(.phi..sub.sh .times.V.sub.shhr) 
But the volume of laminated shale is not yet known for the high resolution 
log. Therefore, substituting equation 19 into 20, one has equation 21: 
EQU .phi..sub.ehr =.phi..sub.ts -V.sub.shlhr .times.(.phi..sub.ts 
-.phi..sub.sh)-(.phi..sub.sh .times.V.sub.shhr) 
One can also write equation 22, which is similar to equation 10: 
##EQU9## 
Substitution of .phi..sub.ehr from equation 21 into equation 22 gives an 
equation 23 for computing volume of laminated shale at the high resolution 
level: 
##EQU10## 
Now, total and effective porosities at the high resolution level are found 
from equations 19 and 20. Volume of dispersed clay is determined from 
equation 24: 
EQU V.sub.cldhr =V.sub.shhr -V.sub.shlhr (24) 
The volume of laminated clay is determined as from equation 25: 
EQU V.sub.cllhr =CLSH.times.V.sub.shlhr (25) 
RESISTIVITY 
In thinly bedded laminated reservoirs, the thickness of the sand and shale 
laminae is usually considerably less than the vertical resolution of a 
conductivity (resistivity) measuring tool. Thus the measured conductivity 
in such conditions is a composite value of sand and shale conductivities 
as defined by equation 1. Geometric analysis of the measured conductivity 
can be presented according to the geometric factor theory as equation 26: 
##EQU11## 
where G.sub.i and C.sub.i are respectively the vertical geometric factors 
and conductivities of thin cross sections above and below of the tool 
measure point. 
Combining equations 1 and 26, one obtains equation 27: 
##EQU12## 
Assuming either that conductivity of either the sand or the shale does not 
change significantly, or that they change linearly within the vertical 
resolution of the induction tool, equation 27 simplifies to equation 28: 
##EQU13## 
Then the conductivity of the sand component can be found by rearranging 
terms to yield equation 29: 
##EQU14## 
Finally, the reconstructed composite conductivity at the high resolution 
level is given by equation 30: 
EQU C.sub.trhr =C.sub.sd .times.(1-V.sub.shlhr)+C.sub.sh .times.V.sub.shlhr(30) 
Recalling that conductivity and .resistivity are inversely related, 
reconstructed high resolution composite resistivity R.sub.trhr is derived 
from C.sub.trhr and is used to determine water saturation at the high 
resolution level. 
WATER SATURATION 
Determination of water saturation in laminated shaly sand reservoirs 
requires an equation permitting both laminated and dispersed clays to be 
present. 
Contrary to this need, the overwhelming majority of water saturation 
equations are developed for reservoir with dispersed clay only as 
exemplified by Worthington, P. F., The Evaluation of Shaly-Sand Concepts 
in Reservoir Evaluation. The Log Analyst, January-February, 1985. The well 
known Poupon equation, Poupon, A., Loy, M. E. and Tixier, M. P., A 
contribution to Electrical Log Interpretation in Shaly Sand, Trans. AIME 
201, 138-145 for laminated reservoir assumes clean sand beds in laminated 
reservoirs. It would be desirable to use the Waxman-Smits model Waxman, M. 
H., Smits, L. J. M., Electrical Conductivities in Oil-Bearing Shaly Sands, 
SPE Journal, June 1968 for water saturation determination because it has a 
strong theoretical and experimental basis and is widely accepted by the 
industry. But the Waxman-Smits equation 31, supra., was developed for 
reservoirs with dispersed clay only: 
##EQU15## 
The following analysis of the Waxman-Smits model shows that it can be 
adjusted for reservoirs which contain both laminated and dispersed clays 
if the parameters are adapted and used accordingly. 
FORMATION RESISTIVITY FACTOR 
The Waxman-Smits model is based on two basic assumptions. The first is a 
parallel conductance mechanism for free electrolyte and clay exchange 
cation components, Waxman, supra. Since the electrolyte contained in the 
pores of shale laminae is customarily called "bound water", in laminated 
reservoirs the parallel conductance mechanism can be assumed for clay 
exchange cations and electrolyte (free or bound) contained in all pores 
whether sand or shale. Therefore, the first assumption is applicable to a 
laminated reservoir in no less degree than to a reservoir with dispersed 
clay. 
The second assumption is that "the electric current transported by 
counterions associated with the clay travels along the same tortuous path 
as the current attributed to the ions in the pore water", Waxman, supra. 
The same geometric factors are attributed to both conductive elements. 
In laminated reservoirs there are interbedded sands and shales which 
usually provide different tortuous paths for the electric current. But 
according to the second assumption, in each of these beds the current 
transported by the counterions should follow the same tortuous path as the 
current transported by the ions in the pore water of the sand or shale 
beds. The resulting heterogeneous conductive media of laminated reservoirs 
should also have the same tortuous paths for currents transported both by 
ions of pore water in sand and shale laminae, and also by counterions 
associated with clay in these beds. Thus the second assumption is 
confirmed for the Waxman-Smits model so that it also applies to laminated 
reservoirs. 
Waxman-Smits experimental data, Waxman, supra. indicates that the formation 
resistivity factor is usually greater in reservoirs with the greater clay 
content assuming their porosities are approximately equal, see the curves 
32 and 34 in FIG. 4. Both samples 1 and 23 are from Waxman supra., Table 7 
and have porosities of about 24%, but the sample 23 has a high Clay 
content (Q.sub.v =1.04), and F*=27 while the sample 1 has F*=12. Shaly and 
clean sands can have the same formation resistivity factor exemplified by 
the curves of FIG. 5, derived also from Waxman, supra., Table 7, as this 
drawing shows, but the relatively clean sample represented by the curve 36 
has considerably lower porosity of 11% than the shaly sample represented 
by the curve 38, or about 23%. 
As stated in Winsauer, W. O., Shearin Jr., H. M., Masson, P. H., and 
Williams, M., Resistivity of Brine-Saturated Sands in Relation to Pore 
Geometry, Bull. AAPG, Vol. 36 (2), February 1952, the formation 
resistivity factor depends on porosity and tortuosity. The Archie equation 
and other formulae which relate formation resistivity to porosity alone 
are mainly approximations which only work well in clean reservoirs where 
tortuosity mainly depends on porosity. Thus the larger formation 
resistivity factors in shalier reservoirs (relative to cleaner reservoirs 
of the same porosity) can be explained by greater tortuosity in shalier 
reservoirs and shales. It has been observed by Parkhomenko, E. I., 
Electrical Properties of Rocks Plenum Press, New York, 1967, that the 
tortuosity of sedimentary rocks increases with decreasing grain size. This 
is not very noticeable in sands because the difference in grain sizes of 
coarse versus fine grained sands is not great when compared to grain sizes 
of clays (generally less than 2 .mu.m), or about two orders of magnitude 
smaller than fine sand grains. 
The formation resistivity factor dependence on the volume of shale is also 
seen from the relationship between the cementation exponent m and CEC or 
Q.sub.v derived from experimental data tabulated in Waxman, supra. The 
exponent m increases in a non-linear fashion from about 1.78 in clean sand 
to about 2.5 in very shaly sand with CEC of 20 meq/100 g and Q.sub.v =1.6 
meq/ml. These are shown in FIGS. 6 and 7 respectively. The dispersion of 
points on the plot m=f(CEC) is considerably less than for the plot of 
m=f(Q.sub.v). This suggests the greater desirability of using the 
relationship between m and CEC for the m computation. Regression analysis 
of this relationship gives equation 32: 
EQU m=1.78+0.0518.times.CEC-0.00072.times.CEC.sup.2 (32) 
Another relationship between m and Q.sub.v is presented in Brown, G. A., 
The Formation Porosity Exponent-The Key to Improved Estimates of Water 
Saturation in Shaly Sands, SPWLA 29th Annual Logging Symposium, Jun. 5-8, 
1988. Thus the formation resistivity factor in the Waxman-Smits equation 
is given by equation 33: 
EQU F*=.phi..sub.t -m (33 
The value of m is preferably determined by equation 32. 
CATION EXCHANGE CAITY AND Q.sub.V 
In the Waxman-Smits model, Q.sub.v is defined as CEC per unit of total pore 
volume. Neither of these parameters can be measured in situ by logging 
tools. They are usually found by core analysis or indirectly from other 
log derived parameters such as volume of shale, SP, GR and others Donovan, 
W. S. and Hilchie, D. W., Natural Gamma Ray Emissions in the Muddy J. 
Formation in Eastern Wyoming, The Log Analyst, vol. 22 (2), March-April, 
1981, Johnson, W., Effect of Shaliness on Log Responses, CWLS Journal, 
June, 1979, and Fertl, W. H. and Frost Jr., E., Evaluation of Shaly 
Clastic Reservoir Rocks, Journal of Petroleum Technology, September 1980. 
A better approach is an empirical equation from Hill, J. J., Shirley, O. 
J., and Klein, G. E., Bound Water in Shaly Sands-Its Relation to Q.sub.v 
and Other Formation Properties, The Log Analyst, May-June 1979, which 
relates Q.sub.v to bound water in clay, total porosity and water 
mineralization of the reservoir. Equation 34 does not depend on conditions 
specific to an area of investigation. 
##EQU16## 
In hydrocarbon bearing reservoirs, ions associated with clay become more 
concentrated in the remaining pore water, Waxman, above, and consequently 
Q.sub.v is divided by S.sub.wt as stated in equation 31. This is the basic 
reason the Waxman-Smits formula has been applicable to shaly reservoirs 
containing only dispersed clay. Indeed, producible hydrocarbons generally 
occur only in sand laminae where they occupy a part of the pore space. 
Shale laminae do not hold producible hydrocarbons; their pore space is 
filled with bound water and no hydrocarbons. Therefore, the concentration 
of ions in shale laminae associated with clay is not dependent on 
hydrocarbon saturation in near-by sand laminae. Consequently, only the 
Q.sub.v related to dispersed clay has to be divided by water saturation in 
sand laminae. For hydrocarbon bearing laminated shaly sand reservoirs, the 
effective concentration of exchange ions Q.sub.v ' is defined as in 
equation 35: 
##EQU17## 
Q.sub.vd and Q.sub.vl are computed by equation 34 using porosities and 
water mineralizations determined for sand and shale laminae. 
Effective water saturation does not change in a sand lamina regardless of 
whether a sand bed is considered separately or as a part of a laminated 
reservoir or a thin laminar layer. Since Waxman-Smits formula computes 
S.sub.wt, S.sub.w can be found from S.sub.wt as shown by Juhasz, I., The 
Central Role of Q.sub.v and Formation Water Salinity in the Evaluation of 
Shaly Formations, SPWLA 20th Annual Logging Symposium, Jun. 3-6, 1979, and 
then S.sub.wts can be determined from S.sub.w by equation 36: 
##EQU18## 
In clear reservoirs filled only with water of a specific mineralization 
generally the higher formation resistivity is associated with higher 
formation resistivity factor. This is not the case in shaly reservoir. 
Compare two shaly sand reservoirs filled with water of the same 
mineralization; one formation with higher formation resistivity factor as 
defined by the Waxman-Smits model, could have higher, lower, or the same 
resistivity as the other formation. In shaly reservoirs, two factors 
compete with each other, namely, formation resistivity factor versus 
Q.sub.v. 
Higher clay content in a shaly reservoir leads to greater formation 
resistivity factor which should result in higher resistivity of the 
reservoir. But at the same time, higher clay content results in greater 
Q.sub.v which tends to reduce the resistivity of the reservoir. These two 
competing forces will prevail differently dependent on factors such as 
clay type, the mode of clay distribution, and the total and effective 
porosities. All of these factors to a certain extent determine values of F 
and Q.sub.v. Water mineralization and water saturation also play very 
important roles. 
As depicted in FIG. 4 where two samples have the same porosity, for all 
water conductivities above 33 mmho/cm, the sample with greater shale 
content has lower conductivity than the clean sample. But the shaly sample 
has higher conductivity for water conductivities below this value of 33 
mmho/cm. In other words at lower water conductivities, Q.sub.v prevails 
over F* in the shaly sample. Indeed, if F* and Q.sub.v of two shaly water 
bearing reservoirs are known, equation 37 computes water conductivity at 
which their conductivities are equal: 
##EQU19## 
This C.sub.w defines crossover points of the function C.sub.t =f(C.sub.w) 
for both shaly reservoirs. 
The presence of hydrocarbons adds a new dimension to the phenomena 
described above. Hydrocarbons increase reservoir resistivity but 
simultaneously increase the Q.sub.vd contribution to the reservoir 
conductivity according to equation 35. Assuming for simplicity that some 
shaly reservoirs have only dispersed clay, water conductivity C.sub.w for 
the two hydrocarbon bearing shaly reservoirs with equal conductivities is 
given by equation 38: 
##EQU20## 
Assume that both reservoirs also have the same water saturation S.sub.w, 
then water conductivity C.sub.w at which their conductivities are equal is 
given by equation 39: 
##EQU21## 
Comparing equations 37 and 39, one observes that a water conductivity at 
which shaly sand beds have the same conductivity is 1/S.sub.wt times 
greater for hydrocarbon bearing reservoir in relation to similar water 
bearing reservoirs. For example, the water bearing samples in FIG. 3 have 
the same conductivity at C.sub.w : 33 mmho/cm, but if they were oil 
bearing with S.sub.wt : 50%, they would have the same conductivity at 
C.sub.w =66 mmho/cm; the conductivities of the shaly sample would be 
greater than those of the clean sample for all water conductivities below 
66 mmho/cm. 
Consequently, water bearing shaly sand reservoirs and shales could have 
higher resistivities than water bearing clean reservoirs with the same 
porosity and water conductivity, but the same hydrocarbon bearing shaly 
reservoirs could have lower resistivity than clean hydrocarbon bearing 
reservoirs with similar water saturation. That is one reason it is so 
important to take into account the increased contribution of clay cation 
exchange capacity to the conductivity of hydrocarbon bearing reservoirs, 
as stipulated by the Waxman-Smits model. 
FORMATION WATER RESISTIVITY 
In the original Waxman-Smits equation, water resistivity does not depend on 
clay content, i.e. it is assumed that the water in clay and sand pores has 
the same mineralization. This is another reason why Waxman-Smits model is 
primarily applicable only to reservoirs with dispersed clay. Since 
dispersed clay is mainly formed after the sand deposition, it is 
reasonable to assume that water mineralization in dispersed clay and sand 
pores is the same. But water mineralization in shales often is less than 
in adjacent shaly sands. Laboratory studies indicate that during 
diagenesis the expelled water shows progressively decreased mineralization 
with increasing overburden pressure, Chilingar, G. V., Rieke, H. H., III, 
and Robertson, J. O., Jr., Relationship Between High Overburden and 
Moisture Content of Hallousite and Dickite Clays, Geol. Soc. Am. Bull, 74, 
1963, and Fertl, W. H., and Timko, D. J., Association of Salinity 
Variations and Geopressures in Soft and Hard Rocks, 11th Prof. Log 
Analysts Symposium, Los Angeles, California, May 1970. Thus the remaining 
water in shales should be fresher (i.e., less mineralized) than in 
surrounding shaly sands because, in the shaly sands, it is better 
protected from overburden compaction by larger sand grains. 
Therefore, to use the Waxman-Smits model for laminated reservoirs, water 
resistivity has to be determined separately for shaly sands and shales. 
Then compound water resistivity can be computed using the percentages of 
shale and sand laminae in the reservoir and the water saturation in the 
shaly sand laminae. Consequently the compound water conductivity is 
computed by equation 40: 
##EQU22## 
Then compound water resistivity is given by equation 41: 
##EQU23## 
EQUIVALENT CONDUCTANCE OF CLAY EXCHANGE CATIONS 
Generally, the equivalent conductance of clay exchange cations or B is a 
function of temperature but for dilute solutions (R.sub.w &gt;0.05 ohmm), it 
also depends on formation water salinity, Waxman supra. Thus if water 
conductivities of sand and shale laminae are different and at least one of 
them is in the dilute solution range, then the equivalent conductances of 
counterions in sand and shale laminae are also different. In this case, 
each should be determined separately from relationship between B and 
R.sub.w at various temperatures presented in Waxman, M. H., and Thomas, E. 
C., Electrical Conductivities in Shaly Sands-I. The Relation Between 
Hydrocarbon Saturation and Resistivity Index; II. The Temperature 
Coefficient of Electrical Conductivity. Journal of Petroleum Technology, 
February 1974. Then conductivity of clay counterions is given by equation 
42: 
##EQU24## 
From this expression and equation 35, the compound equivalent conductance 
of clay exchange cations Bc can be computed by equation 43: 
##EQU25## 
The above discussion leads to the following modified Waxman Smits formula 
for water saturation in the laminated shaly sand reservoirs, see equation 
44: 
##EQU26## 
As before, F* is determined from equations 32 and 33, R.sub.wc is 
determined from equation 41, and B.sub.c is determined from equation 43. 
When using equation 44 for water saturation determination in thinly 
laminated reservoirs, all variables and parameters in the equation have to 
be determined at the high resolution level. 
FIELD EXAMPLE 
Evaluation of thinly laminated shaly sand reservoirs using the technique 
described above has several substantial advantages over conventional 
methods (without reconstructing logs to the high resolution level). The 
present technique provides more accurate and highly differentiated shaly 
sand properties such as shale volume, the mode of clay distribution, 
porosity, and water saturation. One of the examples of log interpretation 
by this technique in comparison with regular log interpretation is 
presented in FIG. 8. For both interpretations, the same parameters and the 
same water saturation equation (the Waxman-Smits model for laminated 
reservoirs) were used. The interval depicted in FIG. 8 represents a thinly 
laminated oil bearing reservoir with maximum porosities of about 33% for 
clean sand beds according to core and log data. Logs used for 
interpretation (FIG. 8a) include density, neutron, one of the dipmeter pad 
conductivities, and deep resistivity. The maximum original resistivity 
R.sub.t is about 2.5 ohmm while reconstructed sand component and composite 
high resolution resistivity logs reach 8 ohmm. At the top of the presented 
interval in FIG. 8 where relatively thick sand beds are developed, the 
high resolution technique computes water saturation only about 10% lower 
than the conventional technique. But in the interval below, where thin 
shaly sand beds prevail, this water saturation increases to 20% or more. 
For example, the thin bed at XX38-XX39 has high resolution water 
saturation only 45% (FIG. 8c) while conventional interpretation gives 
S.sub.w =73% for this bed (FIG. 8b). This difference is almost as large as 
the difference in saturation between hydrocarbon and water producing 
reservoirs. Above (XX29-XX32), two closely located thin shaly sand bends 
(FIG. 8c) are presented by conventional interpretation technique (FIG. 8b) 
as one relatively thick bed with maximum porosity and hydrocarbon 
saturation at XX30.5; according to the present high resolution technique, 
this is the location of the intervening shale lamina. The more accurate 
location of the thin shaly sand beds is another important advantage of the 
high resolution interpretation technique. 
FIG. 9 represents one application for logging a laminate formation. In FIG. 
9 of the drawings, a well 40 passes through a formation of interest at 42. 
Assume, for purposes of description, that the formation 42 is a plurality 
of laminar layers of sand and shale somewhat in the fashion represented in 
FIG. 1a. Assume further that a sonde 44 investigates the formations along 
the well borehole 40, and further assume that data is obtained of some 
property or variable X relating to the formations including the formation 
42. Assume further that the sonde provides data from a low resolution 
investigation, or a high resolution investigation or with some other 
resolution therebetween, all as exemplified in the table given herein. 
This data is output on a logging cable 46 which extends to the surface and 
passes over a sheave 48. The cable 46 is spooled on a storage reel or drum 
50. The conductors are output from the logging cable 46 to a CPU 52. The 
data is prepared in accordance with the present disclosure and is provided 
to a recorder 54. The recorder records the data as a function of depth. 
This is input through an electrical or mechanical depth recording 
mechanism 56. 
In accordance with the teachings of the present disclosure, the variable X 
logged for the formation 42 is processed as taught herein and is converted 
to high resolution data for the variable X. This, as mentioned, can 
encompass several different variables which are measured along the well 
borehole and which variables are exemplified in FIG. 8 of the drawings. 
Referring back to FIG. 8, it is noted that the several traces are best 
identified by the reference numerals placed on FIG. 8 where the reference 
numerals 1-16 identify the following curves or readings: 
1. Caliper 
2. SP 
3. Neutron log 
4. Density log 
5. Dipmeter conductivity 
6. R.sub.t 
7. Reconstructed sand component resistivity 
8. Reconstructed high resolution resistivity 
9. Permeability 
10. Gas indicator 
11. Water saturation 
12. Effective porosity 
13. Oil filled effective porosity 
14. Dispersed clay 
15. Laminated shale 
16. Sand matrix 
While the foregoing is directed to the preferred embodiment, the scope is 
determined by the claims which follow.