Apparatus and method for determining characteristics of subsurface formations

The determination of a "composite" parameter of the formation water in formations surrounding a borehole, for example the composite conductivity of the formation water, is used in the disclosure to obtain a relatively accurate determination of formation characteristics, such as water saturation. The determined values are meaningful even in shaly regions of the formations. In contrast to past approaches which attempted to determine the volume and distribution type of shale or clay present in the formations and then introduce appropriate factors which often involve substantial guesswork, the disclosed technique determines a composite water parameter, for example a composite water conductivity, which represents the conductivity of the bulk water in the formations, including both free water and bound water. Bound water trapped in shales is accounted for in this determination so unlike prior techniques, the shales can be considered as having a porosity. Having determined the composite water conductivity, water saturation can be directly obtained using relatively straightforward relationships which do not require estimates of the volume of shale in the formations. Shale effects are accounted for by the different conductivities (or other parameter such as capture cross sections) of the formation water constituents (free and bound) which make up the total water.

This invention relates to an apparatus and method for investigating 
subsurface formations and, more particularly, to an apparatus and method 
for determining a composite parameter of the formation water in formations 
surrounding a borehole, for example the composite conductivity of the 
formation water. Using the composite parameter, other useful information, 
for example a determination of water saturation, can be accurately made, 
even in shaly formations. 
The amount of oil or gas contained in a unit volume of a subsurface 
reservoir is a product of its porosity and its hydrocarbon saturation. The 
total porosity of a formation, designated .phi..sub.t, is the fraction of 
the formation unit volume occupied by pore spaces. Hydrocarbon saturation, 
designated S.sub.h, is the fraction of the pore volume filled with 
hydrocarbons. In addition to the porosity and hydrocarbon saturation, two 
other factors are necessary to determine whether a reservoir has 
commercial potential; viz., the volume of the reservoir and its 
producibility. In evaluating producibility, it is important to know how 
easily fluid can flow through the pore system. This depends upon the 
manner in which the pores are interconnected and is a property known as 
permeability. 
To determine the amount of producible hydrocarbons in a formation, it is 
useful to obtain a measure of the bulk volume fraction of hydrocarbons 
displaced in invasion of the drilling mud during the drilling operation. 
During drilling, the mud in the borehole is usually conditioned so that 
the hydrostatic pressure of the mud column is greater than the pore 
pressure of the formations. The differential pressure forces mud filtrate 
into the permeable formations. Very close to the borehole, virtually all 
of the formation water and some of the formation hydrocarbons, if present, 
are flushed away by the mud filtrate. This region is known as the "flushed 
zone". The bulk volume fraction of hydrocarbons displaced by invasion in 
the flushed zone is an indication of the amount of "movable" hydrocarbons 
in the particular portion of the formations. This bulk volume fraction of 
the hydrocarbons displaced by invasion can be expressed as .phi..sub.t 
(S.sub.h -S.sub.hr), where S.sub.hr is the residual hydrocarbon saturation 
in the flushed zone (i.e., the saturation of hydrocarbons which were not 
flushed away by the mud filtrate and generally considered as immovable). 
The saturation of the mud filtrate, designated as S.sub.xo, can be 
represented as 
EQU S.sub.xo =(1-S.sub.hr) (1) 
The saturation of hydrocarbons in the uninvaded formations, designated 
S.sub.h, can be expressed as 
EQU S.sub.h =(1-S.sub.w) (2) 
where S.sub.w is the water saturation of the formations; i.e., the fraction 
of the pore spaces filled with water. From the equations (1) and (2), it 
can be seen that the previously set forth expression for the bulk volume 
fraction of oil displaced by invasion, .phi..sub.t (S.sub.h -S.sub.hr), 
can be expressed as 
EQU .phi..sub.t (S.sub.h -S.sub.hr)=.phi..sub.t (S.sub.xo -S.sub.w) (3) 
Generally, relatively accurate determinations of .phi..sub.t can be 
obtained using known logging techniques, so accurate determinations of 
S.sub.xo and S.sub.w are highly useful, inter alia, for determining the 
bulk volume fraction of hydrocarbons displaced by invasion and, therefore, 
the fraction of producible hydrocarbons for particular formations 
surrounding the borehole. 
Classical prior art techniques exist for determining water saturation 
and/or related parameters. It has been established that the resistivity of 
a clean formation (i.e., one containing no appreciable amount of clay), 
fully saturated with water, is proportional to the resistivity of the 
water. The constant of proportionality, designated F, is called the 
formation factor. Thus we have 
EQU F=R.sub.o /R.sub.w ( 4) 
where R.sub.o is the resistivity of the formation 100% saturated with water 
of resistivity R.sub.w. Formation factor is a function of porosity, and 
can be expressed as 
EQU F=a/.phi..sub.t m (5) 
where a and m are generally taken to be 1 and 2, respectively. Using these 
values, the true resistivity, designated R.sub.t, of a clean formation 
containing hydrocarbons is expressed as 
EQU R.sub.t =R.sub.w /S.sub.w.sup.n .phi..sub.t.sup.2 ( 6) 
where n, the saturation exponent, is generally taken to be 2. Using the 
classical equation set forth, one conventional prior art technique 
computes a value, designated R.sub.o ' which is a computed "wet" 
resistivity value and assumes that the formation is fully saturated with 
water; i.e., S.sub.w =1. From relationship (6), it can be seen that 
EQU R.sub.o '=R.sub.w /.phi..sub.t.sup.2 ( 7) 
In this computation, .phi..sub.t may be obtained from logging information, 
for example from neutron and/or density log readings, and R.sub.w may be 
obtained from local knowledge of connate water resistivity or, for 
example, from a clean water-bearing section of a resistivity log. The 
computed value of R.sub.o ' is compared with a measured value of 
resistivity, designated R.sub.t, obtained, for example, from a deep 
investigation resistivity or induction log. In clean zones having no 
hydrocarbons R.sub.o ' will track R.sub.t, but when R.sub.o ' is less than 
R.sub.t, there is an indication of the presence of hydrocarbons. Thus, by 
overlying the computed wet resistivity (R.sub.o ') and the measured 
resistivity (R.sub.t), potential hydrocarbon bearing zones can be 
identified. From equations (6) and (7), it is seen that another way of 
using this information is to obtain a computed value of apparent water 
saturation, designated S.sub.w ', from the relationship 
##EQU1## 
Substantial deviations of S.sub.w ' from unity also indicate potential 
hydrocarbon bearing zones. 
The described types of techniques are effective in relatively clean 
formations, but in shaly formations the shales contribute to the 
conductivity, and the usual resistivity relationships, as set forth, do 
not apply. Accordingly, and for example, the previously described overlay 
or R.sub.o ' and R.sub.t can lead to incorrect conclusions in a shale 
section of the formations, and the overlay in these sections (as well as 
the determination of water saturation taken therefrom) is generally, of 
necessity, ignored. In addition to the results being less useful than they 
might be, this consequence can tend to diminish the credibility of the 
entire computed log comparison and is a disadvantage when attempting to 
commercially exploit the resultant information. Accurate determination of 
S.sub.w can also be difficult in shaly sections. Of course, these are just 
limited examples of how shaliness can interfere with measurement 
interpretation, but similar problems with shaliness arise in other 
situations, such as when invaded zone characteristics (like S.sub.xo) are 
to be determined or when interrupting readings from thermal decay time 
logs in cased boreholes. 
A number of techniques, of varying complexity, are in existence for aiding 
in the interpretation of results obtained in shaly formations. The manner 
in which shaliness affects a log reading depends on the proportion of 
shale and its physical properties. It may also depend upon the way the 
shale is distributed in the formations. It is generally believed that the 
shaly material is distributed in shaly sands in three possible ways; i.e., 
"laminar shale" where the shale exists in the form of laminae between 
which are layers of sand, "structural shale" where the shale exists as 
grains or nodules in the formation matrix, and "dispersed shale" where the 
shaly material is dispersed throughout the sand partially filling the 
intergranular interstices. Shaly-sand evaluations are typically made by 
assuming a particular type of shale distribution model and incorporating 
into the model information which indicates the volume of shale or the 
like. For example, in a laminated sand-shale simplified model, an equation 
of the form of equation (6) is set forth, but includes a second term which 
is a function of the bulk-volume fraction of shale in the laminae. The 
same is true for another known model wherein a term is developed which 
depends upon the volume fraction of shale as determined from a total clay 
indicator. In a dispersed shale simplified model, values are developed for 
an "intermatrix porosity" which includes all the space occupied by fluids 
and dispersed shale and another value is developed representing the 
fraction of that porosity occupied by the shale. Still another approach 
relates the conductivity contribution of the shale to its cation exchange 
capacity, this capacity being determined, inter alia, from the volume of 
clay. 
The described prior art techniques, which require either a determination of 
the volume of shale or clay, or similar information, have been 
satisfactory in some applications. However, in addition to the difficulty 
of accurately obtaining information concerning the volume and composition 
of shale or clay and its conductivity, a further problem with prior art 
simplified models is that various forms of shale may occur simultaneously 
in the same formation. Reliable techniques, some of which use extensive 
statistical treatment of data, do exist and generally yield good results, 
but tend to be relatively complex and may require either powerful 
computing equipment and/or substantial processing time. 
It is one object of the present invention to provide a solution to the 
indicated prior art problems and to set forth techniques which are 
effective even in shaly formations, but which are not unduly complex or 
difficult to implement. 
SUMMARY OF THE INVENTION 
Applicant has discovered that determination of a "composite" parameter of 
the formation water in formations surrounding a borehole, for example the 
composite conductivity of the formation water, allows a relatively 
accurate determination of formation characteristics, such as water 
saturation, the determined values being meaningful even in shaly regions 
of the formations. In contrast to past approaches which attempted to 
determine the volume of shale or clay present in the formations and then 
introduce appropriate factors which often involve substantial guesswork, 
applicants' technique determines a composite water parameter, for example 
a composite water conductivity, which represents the conductivity of the 
bulk water in the formations, including both free water and bound water. 
Bound water trapped in shales is accounted for in this determination, so 
unlike prior techniques, the shales can be considered as having a 
porosity. Having determined, at each depth level, the composite water 
conductivity, water saturation can be directly obtained using relatively 
straightforward relationships which do not require estimates of the volume 
of shale in the formations. Shale effects are accounted for in the present 
invention by the different conductivities (or other parameter such as 
capture cross sections) of the formation water constituents (free and 
bound) which make up the total water. As used herein, "free water" is 
generally intended to mean water that is reasonably free to be moved under 
normal reservoir dynamics, whereas "bound water" is generally intended to 
mean water that is not reasonably free to be moved under normal reservoir 
dynamics. 
In accordance with an embodiment of the invention, there is provided an 
apparatus for determining, at each depth level, a composite parameter 
(such as the composite conductivity or the composite capture cross 
section) of the formation water in formations surrounding a borehole. 
Means are provided for deriving a first quantity representative of the 
parameter attributable to the free water in the formations. Means are also 
provided for deriving a second quantity representative of the fraction of 
bound water in the formations. (As will become clear, the second quantity 
could alternatively be obtained indirectly from the fraction of free 
water). Further means are provided for deriving a third quantity 
representative of the parameter attributable to the bound water in the 
formations. The composite parameter is then determined as a function of 
the first, second and third quantities. 
In one form of the invention, a fourth quantity is derived, as the 
difference between the third and first quantities. The composite parameter 
is then determined as the sum of the first quantity and the product of the 
second and fourth quantites. 
In an embodiment of the present invention the composite water conductivity, 
designated .sigma..sub.wc ', is expressed by the following relationship: 
##EQU2## 
where .sigma..sub.wf is the conductivity of the free water in the 
formations, .sigma..sub.wb is the conductivity of the bound water in the 
formations, S.sub.w is the water saturation of the formations (which 
equals .phi..sub.w /.phi..sub.t), and S.sub.wb is the saturation of the 
bound water in the formations (which equals .phi..sub.wb /.phi..sub.t). 
The expression (9) apportions the composite water conductivity as between 
the conductivity of the free water (the above-indicated first quantity) 
and the conductivity of a difference term which expresses the difference 
between the conductivities of the bound water and the free water (the 
above-indicated fourth quantity). Mathematical manipulation shows that 
another form of expression (9) is 
##EQU3## 
In this form, the composite water conductivity can be viewed as the sum of 
a first term, which represents the fraction of free water times the 
conductivity of the free water, plus a second term which represents the 
fraction of bound water times the conductivity of the bound water. As 
implied above, the fraction of free water, S.sub.wf /S.sub.w, (which is 
the unity complement of the bound water fraction--since the total water 
volume consists of the free water volume plus the bound water volume) 
could alternately be used in expressions (9) or (10). For example, the 
form of expression (10) would then be 
##EQU4## 
which can be seen to be equivalent to (10) since S.sub.w =S.sub.wf 
+S.sub.wb. Accordingly, when the term "fraction of bound water", or the 
like, is used in this context, it will be understood that its complement 
(the fraction of free water) could alternatively be employed in 
appropriate form. 
In another embodiment of the invention, the composite parameter of the 
formation water is the composite water capture cross section, designated 
.SIGMA..sub.wc '. As is shown in the art, capture cross section is a 
measure of the fraction of thermal neutrons absorbed per unit time, and is 
typically measured using a thermal neutron decay time ("NDT") logging 
device of the type described, for example, in U.S. Pat. No. RE 28,477. The 
composite water capture cross section, .SIGMA..sub.wc ', is expressed 
herein as 
##EQU5## 
which is similar to expression (9), but where .SIGMA..sub.wf is the 
capture cross section of the free water in the formations and 
.SIGMA..sub.wb is the capture cross section of the bound water in the 
formations. 
In accordance with a further feature of the invention, a value of water 
saturation is generated and provides meaningful information even in shaly 
regions. This obviates the prior art technique of estimating an 
appropriate "cementation" exponent for shaly formations. 
In accordance with still further features of the invention, relationships 
similar to (9) or (10) can be set forth in terms of a generalized 
parameter, "P", and utilized to obtain a free, a bound, or a composite 
water parameter, depending on what information is desired and what 
information is measurable or deriveable. In particular, if it is desired 
to obtain a parameter of the free water, one can set forth the following 
generalized relationship which is similar in form to relationship (9) 
above 
##EQU6## 
where P.sub.wc is a composite water parameter, P.sub.wb is a bound water 
parameter, and P.sub.wf ' is the free water parameter to be determined. In 
an embodiment of the invention, the free water parameter to be determined 
is in the form of a variable .phi..sub.w .alpha..sub.wf, defined as the 
signal attenuation attributable to formations when assuming that 
substantially all of the water therein is free water. Means are provided 
for deriving a function representative of the parameter (attenuation in 
this case) in at least one region of the formations (typically a clean 
sand region) in which substantially all of the water present is free 
water. Means are also provided for deriving a quantity representative of 
water content in the formations surrounding a particular depth location in 
the borehole. This quantity may be t.sub.pl, the travel time of microwave 
electromagnetic energy in the formations, which is dependent on water 
content. The free water parameter (in the form of the variable .phi..sub.w 
.alpha..sub.wf in this case) at the particular depth level is then 
determined from the derived function and the water content representative 
quantity. Measurements of attenuation and travel time are typically 
obtained using an "EMP" microwave electromagnetic propagation logging 
device. 
In terms of the attenuation, .alpha., the relationship (9a) can be 
expressed as 
##EQU7## 
where .alpha..sub.wb is the bound water conterpart of .alpha..sub.wf, and 
.alpha..sub.wc is a "composite" attenuation for the actual formation 
water. 
As will be described further hereinbelow, the "apportionment" of 
attenuation, as between the free end bound water which is indicated by 
expression (9b) leads to a technique for determining the fraction of bound 
water, S.sub.wb /S.sub.w once the values of .alpha., .alpha..sub.wf and 
.alpha..sub.wb have been established. In particular, S.sub.wb /S.sub.w can 
be determined from 
##EQU8## 
which follows directly from relationship (9b). 
Further features and advantages of the invention will become more readily 
apparent from the following detailed description when taken in conjunction 
with the accompanying drawings.

DESCRIPTION OF THE PREFERRED EMBODIMENT 
Referring to FIG. 1, there is shown a representative embodiment of an 
apparatus in accordance with the present invention for investigating 
subsurface formations 31 traversed by a borehole 32. The borehole 32 is 
typically filled with a drilling fluid or mud which contains finely 
divided solids in suspension. The investigating apparatus or logging 
device 40 is suspended in the borehole 32 on an armored cable 33, the 
length of which substantially determines the relative depth of the device 
40. The cable length is controlled by suitable means at the surface such 
as a drum and winch mechanism (not shown). Circuitry 51, shown at the 
surface, although portions thereof may typically be downhole, represents 
the overall processing circuitry for the various logging units of 
apparatus 40. 
The investigating apparatus 40 includes a suitable resistivity-determining 
device such as an induction logging device 41. As is shown in the art, 
formation resistivity or conductivity is indicated by the induction log 
readings, the measured conductivity being designated as .sigma..sub.t. The 
downhole investigating apparatus also includes a sidewall epithermal 
neutron exploring device 42 having a source and detector mounted on a skid 
42A. A device of this type is disclosed, for example, in U.S. Pat. No. 
2,769,918. Each count registered in the epithermal neutron detector is 
received by a processing circuit in the overall circuitry 51 which 
includes a function former that operates in well known manner to produce a 
signal .phi..sub.N which represents the formation porosity as determined 
by the neutron logging device. The investigating apparatus 40 further 
includes a formation density exploring device 43 for producing well 
logging measurements which can be utilized to calculate the bulk density 
of the adjoining formations, in known manner. In this regard, a skid 43A 
houses a source and two detectors (not shown) spaced different differences 
from the source. This arrangement of source and detectors produces signals 
that correspond to the bulk density of the earth formations as is 
described, for example, in the U.S. Pat. No. 3,321,625. The circuitry 51 
includes conventional circuits which convert the signals derived from the 
short and long spacing detectors to a computed bulk density. If desired, a 
caliper signal may also be applied in determining bulk density, as is 
known in the art. The resulting bulk density is applied to porosity 
computing circuitry within the block 51 which computes the porosity, as 
derived from the bulk density, in well known fashion. The derived porosity 
is designated as .phi..sub.D. The investigating apparatus includes a still 
further device 44 which is a gamma ray logging device for measuring the 
natural radioactivity of the formations. The device 44, as known in the 
art, may typically include a detector, for example a gamma ray counter, 
which measures the gamma radiation originating in the formations adjacent 
the detector. An output of circuitry 51 is a signal designated "GR" which 
represents the gamma ray log reading. Further devices may be provided, as 
required in accordance with variations of the invention as described 
hereinbelow. For example, a device 45 is available for obtaining 
measurement of the spontanteous potential ("SP") of the formations. This 
device may be of the type disclosed in U.S. Pat. No. 3,453,530, this 
patent also disclosing deep and shallow resistivity devices. Also, an 
electromagnetic propagation tool ("EMP") 46 is available, and includes a 
pad member 46A that has transmitting and receiving antennas therein. 
Microwave electromagnetic energy is transmitted through the formations 
(typically the invaded zone) and formation characteristics are determined 
by measuring the attenuation and/or phase (or velocity) of received 
microwave energy. This type of logging tool is described in U.S. Pat. No. 
3,944,910. Measurements indicative of attenuation, designated .alpha., and 
of travel time (which depends on velocity), designated t.sub.pl, are 
available from this tool. Also, in the copending U.S. Patent application 
Ser. Nos. 806,983 and 788,393, assigned to the same assignee as the same 
present application, there are disclosed techniques for obtaining an 
"EMP"-derived conductivity measurement, designated .sigma..sub.EMP, and 
for obtaining a measurement of bound water filled porosity, designated 
.phi..sub.wb. Signals representative of these measurement values are 
illustrated as being available outputs of circuitry 51. An NDT device 47, 
for example of the type disclosed in U.S. Pat. No. RE 28,477, is also 
available and results in an output capture cross section value, .SIGMA., 
from processing circuitry 51. 
To keep the investigating apparatus 40 centered in the borehole, extendable 
wall-engaging members 42B, 43B and 46B may be provided opposite the 
members 42A, 43A and 46A. For centering the upper portion of the 
investigating apparatus, centralizers 49 may also be provided. As noted, a 
borehole caliper can be combined with the arms which extend the skids and 
supply a signal representative of borehole diameter to the circuitry 51. 
While all of the measurements to be used in practising the invention are 
shown, for ease of explanation in this illustrative embodiment, as being 
derived from a single exploring device, it will be understood that these 
measurements could typically be derived from a plurality of exploring 
devices which are passed through the borehole at different times. In such 
case, the data from each run can be stored, such as on magnetic tape, for 
subsequent processing consistent with the principles of the invention. 
Also, the data may be derived from a remote location, such as by 
transmission therefrom. 
One or more of the signal outputs of block 51 are illustrated in FIG. 1 as 
being available to computing modules 60, 70, 80 and 510. In the embodiment 
of FIG. 1, the computing module 60 generates a signal representative of an 
apparent composite water conductivity, designated .sigma..sub.wco ', 
consistent with the relationship (9). The computing module 70 is 
responsive to the signal representative of .sigma..sub.wco ', and to the 
signals from block 51 (in particular a porosity-indicative signal), to 
generate a "wet" conductivity signal, .sigma..sub.o '. The computing 
module 80 generates a computed value of water saturation, S.sub.w ', in 
accordance with a relationship to be set forth. The computing module 510 
is utilized in the generation of free and bound water attenuation values 
and a signal representative of the bound water fraction. These signals, 
along with some or all of the outputs of circuitry 51, are recorded as a 
function of depth on recorder 90. 
Referring to FIGS. 2 and 3, there are shown embodiments of the computing 
modules, 60 and 70 of FIG. 1. Initially, structural components of the 
modules will be described. The source of various signals, along with 
further rationale of the configurations, will then be set forth. A pair of 
difference circuits 601 and 602 are provided. The positive input terminal 
of circuit 601 receives the signal GR, i.e., a signal representative of 
the output of the gamma ray logging device 44. The positive input terminal 
of circuit 602 receives a signal designated GR.sub.wb, which is a signal 
level representative of a gamma ray log level for the bound water of the 
formations being investigated. The negative input terminals of both 
difference circuits 601 and 602 receive a signal level designated 
GR.sub.wf, which is a gamma ray log level for the free water in the 
formations being investigated. The outputs of circuit 601 and 602, which 
are respectively GR-GR.sub.wf and GR.sub.wb -GR.sub.wf, are coupled to a 
ratio circuit 603 which produces a signal proportional to the ratio of the 
output of circuit 601 divided by the output of circuit 602. The output of 
ratio circuit 603 is a signal representative of S.sub.wb, i.e., the 
saturation of the bound water of the formations in accordance with the 
relationship 
##EQU9## 
The output of ratio circuit 603 is coupled via limiter 604 to one input to 
a multiplier circuit 605. The other input to multiplier circuit 605 is the 
output of a difference circuit 606. The circuit 606 receives at its 
positive input terminal a signal level representative of .sigma..sub.wb, 
i.e. the conductivity of the bound water in the formations being 
investigated. The negative input terminal of difference circuit 606 
receives a signal level representative of .sigma..sub.wf, i.e. the 
conductivity of the free water of the formations. This latter signal is 
also one input to a summing circuit 607 whose other input is the output of 
multiplier circuit 605. The output of summing circuit 607 is a signal 
representative of the apparent composite water conductivity of the 
formations being investigated, i.e. 
EQU .sigma..sub.wco '=.sigma..sub.wf +S.sub.wb (.sigma..sub.wb -.sigma..sub.wf) 
(13) 
This expression is seen to be the same as the expression (9) above for 
composite water conductivity, .sigma..sub.wc, except that S.sub.w is 
assumed to be 1, which means that the result is an "apparent" composite 
water conductivity. 
In FIG. 3 there is shown an implementation of the computing module 70 of 
FIG. 1 which is utilized to generate a signal representative of 
.sigma..sub.o ', i.e. the computed "wet" conductivity of the investigated 
formations. The circuitry 51 (FIG. 1) includes a porosity computing 
circuit 511 which is responsive to the signals representative of 
.phi..sub.N and .phi..sub.D. The circuit 511 uses this information, in 
well known manner, to produce a signal generally known as .phi..sub.ND 
that incorporates information from both the neutron and the density log 
readings to obtain an indication of formation total porosity, designated 
.phi..sub.t. Techniques for obtaining .phi..sub.ND are well known in the 
art, and a suitable neutron-density porosity computing circuit is 
disclosed, for example, in the U.S. Pat. No. 3,590,228 of Burke. It will 
be understood however, that any suitable alternate technique for obtaining 
.phi..sub.t can be employed, including, for example, techniques that use 
other logging information, such as from a sonic log. The output of circuit 
511 is coupled to a squaring circuit 701 whose outut is accordingly 
proportional to .phi..sub.t.sup.2. This signal is, in turn, coupled to one 
input terminal of a multiplier circuit 702, the other input to which is 
.sigma..sub.wco ', i.e. the apparent composite water conductivity as 
determined by computing module 60 (FIG. 1, FIG. 2). Accordingly, the 
output of multiplier circuit 702 (which is also the output of computing 
module 70--FIG. 1), is a signal proportional to .sigma..sub.wco ' 
multiplied by .phi..sup.2, and is thus indicative of the computed "wet" 
conductivity of the formations, .sigma..sub.o ', in accordance with a 
relationship analagous to (7) above; viz.: 
EQU .sigma..sub.o '=.sigma..sub.wco '.phi..sub.t.sup.2 (14) 
The manner in which the inputs to computing module 60 can be developed will 
now be described. In particular, one preferred technique for obtaining 
values of S.sub.wb, .sigma..sub.wb and .sigma..sub.wf is as follows: Log 
values of .sigma..sub.t, GR and .phi..sub.t are initially obtained over a 
depth range of interest. Using the measured conductivity, .sigma..sub.t 
(which is preferably from a deep resistivity measurement), one can 
compute, at each depth level over the range of interest, a value 
designated .sigma..sub.wa ' as 
EQU .sigma..sub.wa '=.sigma..sub.t /.phi..sub.t.sup.2 (15) 
This is similar in form to relationship (7) above, and it is seen that 
.sigma..sub.wa ' is a simple computed apparent water conductivity [not to 
be confused with the apparent composite water conductivity, 
.sigma..sub.wco', developed in accordance with relationship (13)]; that 
is, it is the computed value of water conductivity that would be expected 
in order for the obtained conductivity measurement (.sigma..sub.t) to 
result from the obtained total porosity measurement, assuming that the 
total porosity is water-filled (viz. assuming that S.sub.w =1). Stated 
another way, a formation of porosity .phi..sub.t which is filled with 
water of conductivity .phi..sub.wa ' would (according to the basic Archie 
relationship) result in the measured formation conductivity .sigma..sub.t. 
If desired, a computing circuit of the type employed in FIG. 3 (which uses 
another form of the relationship to develop .sigma..sub.o ' from 
.sigma..sub.wco ') could be utilized to obtain .sigma..sub.wa ' in 
accordance with relationship (15), except that a divider is used instead 
of a multiplier. The inputs to the divider are .sigma..sub.t and 
.sigma..sub.t.sup.2. Having obtained .sigma..sub.wa ' at each depth level 
over the depth range of interest, the inverse of these values can now be 
utilized, in conjunction with gamma ray (GR) log readings taken over the 
same depth range, to generate a frequency cross-plot of the type 
illustrated in FIG. 5. Frequency cross-plots are commonly used in the well 
logging art (see, for example, Schlumberger "Log Interpretation-Volume 
II", 1974 Edition). At each depth level, the values of 1/.sigma..sub.wa ' 
and GR result in a point on the cross-plot. When all points have been 
plotted, the number of points which fall within each small elemental area 
(of a selected size) on the plot are summed and presented numerically. The 
resultant plot is as shown in FIG. 5, with the numbers thereon 
representative of the frequency of occurrence of points at each particular 
elemental area on the plot. In the illustrated example, the region 
designated by enclosure 501 contained the highest concentration of points 
(i.e. more than five points at each elemental area), so the frequencies of 
occurrence within this region are omitted for clarity of illustration. The 
position on the GR axis designated as GR.sub.wf is indicated by the line 
of lowest gamma ray readings on the plot, as shown in dashed line. The 
position on the GR axis designated as GR.sub.wb is indicated by the GR 
value at which increasing GR no longer results in increasing values of 
1/.sigma..sub.wa '. This means that at GR.sub.wb essentially all the water 
in the formations is bound (typically by whatever shaliness in present). 
Any further shaliness or increases in the volume of clay would mean an 
increase in GR toward GR.sub.max (FIG. 5), but would not increase the 
bound water fraction since essentially all water present was indicated as 
bound at the GR.sub.wb line. The fraction of bound water is then 
determined by interpolation between the reference lines GR.sub.wf and 
GR.sub.wb, that is, as 
##EQU10## 
The line on the 1/.sigma..sub.wa ' axis at which 1/.sigma..sub.wa ' no 
longer varies substantially with GR (beyond GR.sub.wb) is indicative of 
1/.sigma..sub.wb, since, as previously noted, at this point on the plot 
essentially all of the formation water is bound. Accordingly 
.sigma..sub.wb is derived from the dashed line labelled with this 
designation. Applicant has found that .sigma..sub.wb is substantially a 
constant and has a value of about 7 mhos/m at 75.degree. C. It is not, 
however, considered a universal constant and may vary somewhat in 
different regions. In any event, it is determinable from e.g. the 
cross-plot of FIG. 5. The value of the free water conductivity 
.sigma..sub.wf, can be obtained, for example, from the free water dashed 
line on the FIG. 5 plot. Alternatively, as is known in the art, 
.sigma..sub.wf can be obtained from a clean sand section of a resistivity 
log or from local knowledge. It will be understood that alternate 
techniques can be utilized to obtain at least some of the values 
considered herein. 
With values of GR.sub.wf, GR.sub.wb, .sigma..sub.wf, and .sigma..sub.wb 
having been established for the depth range of interest, corresponding 
signal levels can be input to the computing module 60 (FIG. 2). Now, log 
values of GR (as a function of depth can be input to module 60 and 
.sigma..sub.wco ' can be output and recorded (if desired) on a dynamic 
basis. At the same time, the computing module 70 (FIG. 3) generates 
.sigma..sub.o ' as an output to recorder 90. This signal can now be 
overlayed with .sigma..sub.t, to great advantage in identifying potential 
hydrocarbon bearing zones. 
FIG. 6 illustrates the nature of the signals which can be recorded by the 
recorder 90 in the embodiment of FIG. 1. The vertical axis represents 
depth. The middle track shows the inverses of .sigma..sub.o ' (dashed 
line) and .sigma..sub.t (solid line); i.e., the computed "wet" resistivity 
and the measured deep resistivity, respectively. The regions of divergence 
of these curves, for example the regions designated 2 and 3, indicate that 
the measured deep resistivity is substantially greater than the computed 
"wet" resistivity (or, conversely, that the measured deep conductivity is 
substantially less than the computed "wet" conductivity), thereby 
indicating that they are potential hydrocarbon bearing zones. The left 
hand track illustrates the output of a spontaneous potential (SP) log over 
the same depth range. Relatively stable values of the SP, for example in 
the regions designated 4 and 5 are at the "shale baseline" and 
characteristic of shaly regions. It is seen that the resistivity curves 
generally track each other even in the shaly zones, as should be the case 
for water-bearing shale regions. This continuous tracking of the measured 
and derived resistivity signals is an important advantage of the present 
invention since comparable prior art techniques are generally unreliable 
in shaly regions, as discussed in the Background section hereof. 
The determination of a computed value of water saturation, S.sub.w ', will 
now be considered. Relation (9) above indicated that the composite water 
conductivity, .sigma..sub.wc, is expressed as: 
##EQU11## 
From equation (6) we can write 
EQU .sigma..sub.t =.sigma..sub.w S.sub.w.sup.2 .phi..sub.t.sup.2 (17) 
where .sigma..sub.w is the (unknown) actual conductivity of the formation 
water. Substituting the expression for composite water conductivity 
(.sigma..sub.wc) for .sigma..sub.w in (17) gives: 
##EQU12## 
The apparent water conductivity .sigma..sub.wa ' (as described in 
conjunction with FIG. 5) is equal to .sigma..sub.t /.phi..sub.t.sup.2. 
Substituting into (18) gives 
EQU .sigma..sub.wa '=S.sub.w.sup.2 .sigma..sub.wf +S.sub.w S.sub.wb 
(.sigma..sub.wb -.sigma..sub.wf) (19) 
which can be rewritten as: 
EQU [.sigma..sub.wf ]S.sub.w.sup.2 +[S.sub.wb (.sigma..sub.wb 
-.sigma..sub.wf)]S.sub.w -.sigma..sub.wa '=0 (20) 
This quadradic equation can be solved for S.sub.w to obtain: 
##EQU13## 
From relationship (21) it is seen that a value of water saturation, 
obtained using the composite (free and bound) water technique of the 
present invention, can provide meaningful information even in shaly 
regions, since the effects of the shales in binding a portion of the 
formation waters is accounted for in the relationship. Accordingly, the 
prior art technique of estimating an appropriate "cementation" exponent 
for shaly formations is obviated. 
FIG. 4 illustrates an implementation of the computing module 80 utilized to 
generate a signal representative of computed water saturation, designated 
S.sub.w ', in accordance with relationship (21). The signal representative 
of "true" or measured resistivity, .sigma..sub.t (FIG. 1), is one input to 
a ratio circuit 811. The other input to ratio circuit is the output of a 
squaring circuit 812 whose input is a signal representative of 
.phi..sub.t. Accordingly, the output of ratio circuit 811 is proportional 
to .sigma..sub.t /.phi..sub.t.sup.2, which equals the apparent formation 
conductivity, .sigma..sub.wa '. This signal is, in turn, coupled as one 
input to a multiplier circuit 805 whose other input is a signal 
representative of .sigma..sub.wf. The output of multiplier 805 is coupled, 
with a weighting factor of 4, to one input of a summing circuit 804. The 
signal .sigma..sub.wf is also coupled to the negative input terminal of a 
difference circuit 801, the positive input terminal of which receives a 
signal representative of .sigma..sub.wb. The output of difference circuit 
801 is one input to a multiplier 802. The other input to multiplier 802 is 
a signal representative of S.sub.wb, which may be derived, for example, 
from the output of the limiter 604 of FIG. 2. Accordingly, the output of 
multiplier 802 is a signal representative of S.sub.wb (.sigma..sub.wb 
-.sigma..sub.wf). This signal is coupled to a squaring circuit 803 and to 
the negative input terminal of a difference circuit 807. The output of 
squaring circuit 803 is coupled to the other input terminal of summing 
circuit 804 whose output is, in turn, coupled to a square root circuit 
806. The output of the square root circuit 806 is coupled to the positive 
input terminal of difference circuit 807. The output of difference circuit 
807 is coupled to one input of a ratio circuit 808, the other input of 
which receives the signal representative of .sigma..sub.wf, this signal 
being afforded a weighting factor of 2. The output of ratio circuit 808 is 
the desired signal representative of S.sub.w ', in accordance with 
relationship (21). The right track of FIG. 6 illustrates the recorded 
values of the computed water saturation, S.sub.w '. 
The determination of a composite conductivity and determination of water 
saturation, in accordance with the principles of the invention, applies 
equally well in the invaded zone of the formations. In the relationships 
(9) and (18) for example, the quantity .sigma..sub.wf would be replaced by 
.sigma..sub.mf (i.e. the conductivity of the invading mud filtrate) and 
the water saturation S.sub.w would be replaced by the invaded zone 
saturation S.sub.xo. The EMP logging device referred to above measures 
characteristics of the invaded zone. In the abovereferenced U.S. Patent 
application Ser. No. 788,393, a technique is disclosed for measuring 
.phi..sub.wb using an EMP logging device. This technique can be utilized 
as an alternate herein for obtaining S.sub.wb from S.sub.wb 
=.sigma..sub.wb /.sigma..sub.t. In another abovereferenced U.S. Patent 
application Ser. No. 806,983 it is disclosed that conductivity as measured 
using an EMP device, and designated .sigma..sub.EMP, is related to the 
conductivity of the formation water, .sigma..sub.w, as a linear function 
of water-filled porosity, .phi..sub.w, i.e.: 
EQU .sigma..sub.EMP =.sigma..sub.w .phi..sub.w (22) 
Since S.sub.w =.phi..sub.w /.phi..sub.t and .phi..sub.w =.phi..sub.t 
S.sub.w, relationship (22) can be expressed as: 
EQU .sigma..sub.EMP =.phi..sub.t S.sub.w .sigma..sub.w (23) 
Substituting the expression (9) composite water conductivity for 
.sigma..sub.w into (23) gives: 
##EQU14## 
Substituting .sigma..sub.mf for .sigma..sub.wf and S.sub.xo ' for S.sub.w 
and solving for S.sub.xo ' yields 
##EQU15## 
Referring to FIG. 10, there is shown a block diagram of a computing module 
80' suitable for obtaining a signal which represents the computed invaded 
zone water saturation, S.sub.xo ', in accordance with relationship (25). A 
ratio circuit 111 receives as one input a signal representative of 
.sigma..sub.EMP, and as its other input a signal representative of 
.phi..sub.t. The signal .sigma..sub.EMP may be derived from the EMP device 
46 (FIG. 1) by using processing circuitry 51 as disclosed, for example, in 
the above referenced copending U.S. Patent application Ser. No. 806,983. 
Another ratio circuit 112 receives as one input a signal representative of 
.phi..sub.wb, and as its other input the signal representative of total 
porosity, .phi..sub.t. As noted just above, .phi..sub.wb can be derived 
from the measurements taken with an EMP logging device and, in this 
example, is utilized, in conjunction with .phi..sub.t, to obtain S.sub.wb 
(the output of ratio circuit 112). It will be understood, however, that 
S.sub.wb can be obtained using alternate techniques, such as those 
described herein. A difference circuit 113 receives as its input the 
signals representative of .sigma..sub.wb (which may be obtained as 
indicated above and is typically, although not necessarily, about 7 mhos/m 
at 75.degree. C.) and .sigma..sub.mf. The outputs of ratio circuit 112 and 
difference circuit 113 are coupled to a multiplier circuit 114 whose 
output is therefore S.sub.wb (.sigma..sub.wb -.sigma..sub.mf). The output 
of ratio circuit 111 and multiplier circuit 114 are coupled to still 
another difference circuit 115. The output of difference circuit 115 is 
therefore seen to represent the numerator in expression (25). This output, 
and the signal representative of .sigma..sub.mf, are the inputs to another 
ratio circuit 116, whose output is seen to be representative of S.sub.xo 
', in accordance with expression (25). This signal can be recorded, in the 
manner of the illustration in FIG. 5. 
The spontaneous potential measurements from SP device 45 (FIG. 1) can also 
be used, for example, as an alternate technique for obtaining values of 
S.sub.wb. The SP measurement can be expressed as 
##EQU16## 
where K is a constant dependent upon absolute temperature and 
.sigma..sub.mf ' is a composite conductivity for the invaded zone mud 
filtrate, similar in form to .sigma..sub.wc ' as expressed by relationship 
(9). Using relationship (9) as a basis, we have: 
EQU S.sub.w .sigma..sub.wc '=S.sub.w .sigma..sub.wf +S.sub.wb (.sigma..sub.wb 
-.sigma..sub.wf) (27) 
and 
EQU S.sub.xo .sigma..sub.mf '=S.sub.xo .sigma..sub.mf +S.sub.wb (.sigma..sub.wb 
-.sigma..sub.mf) (28) 
Substituting (27) and (28) into (26) and rearranging gives: 
##EQU17## 
In a water-bearing region of the formations where S.sub.xo =S.sub.w 
relationship (29) reduces to: 
##EQU18## 
Therefore, the relationship (30) can be utilized (taking SP from a 
water-bearing region) as an alternate technique for obtaining S.sub.wb. 
FIG. 11 illustrates circuitry that can be utilized to obtain a signal 
representative of S.sub.wb in accordance with relationship (30). The 
combination of ratio circuit 121, antilog circuit 122, difference circuit 
124 and multiplier 126 are used to obtain the numerator, while ratio 
circuit 123, antilog circuit 122, and difference circuit 125 are used to 
obtain the denominator of .nu.. The ratio circuit 127 then yields .nu. and 
summing circuit 128 and inverter 129 are used to obtain a signal 
representative of S.sub.wb. 
In the previously described embodiments, the determined composite parameter 
of the formations has been the composite conductivity (or resistivity). 
Another composite parameter which can be determined is the composite 
capture cross section, as obtained using an NDT log plus inputs 
corresponding to those indicated above. As is well known, the NDT is 
particularly useful in cased holes where resistivity logs cannot be used. 
In such case, the relationship (11) as set forth above is: 
##EQU19## 
An apparent composite capture cross section, designated .SIGMA..sub.wco ', 
can be obtained in the same manner that .sigma..sub.wco ' was developed 
above, and by using the computing module 60' illustrated in FIG. 7. In 
FIG. 7, the multiplier 705, difference circuit 706, and summing circuit 
707 operate in the same fashion as the corresponding units 605, 606 and 
607 of FIG. 2. Suitable values of .SIGMA..sub.wf, .SIGMA..sub.wb and 
S.sub.wb can be obtained by cross-plotting .SIGMA. against GR in the 
manner described in conjunction with FIG. 5. The only difference is that 
instead of using relationship (15) to obtain a computed apparent water 
conductivity, an apparent water capture cross section, .SIGMA..sub.wa ', 
to be plotted against GR, is obtained from the known relationship 
##EQU20## 
where .SIGMA..sub.ma is the matrix capture cross section for the 
particular lithology encountered. The circuitry of FIG. 8, including 
difference circuit 881, ratio circuit 882 and summing circuit 883, can be 
employed to obtain .SIGMA..sub.wa ' in accordance with relationship (31). 
After plotting .SIGMA..sub.wa ' against GR, .SIGMA..sub.wf and 
.SIGMA..sub.wb can be determined, for example, as indicated in conjunction 
with FIG. 5. S.sub.wb can be obtained using the arrangement of circuits 
601, 602, 603, 604, of FIG. 2, as described in conjunction therewith. 
Having determined .SIGMA..sub.wco ', one can now compute a "wet" capture 
cross section (analagous to .sigma..sub.o ' obtained using relationship 
(14) above) from: 
EQU .SIGMA..sub.o '=.phi..sub.t .SIGMA..sub.wco '+(1-.phi..sub.t).SIGMA..sub.ma 
(32) 
The circuitry of FIG. 9, including difference circuit 901, multipliers 902 
and 903, and summing circuit 904, can be utilized to generate a signal 
representative of .SIGMA..sub.o '. This signal can then be overlayed with 
the measured log value, .SIGMA., in the manner illustrated in the central 
rack of FIG. 5, to reveal potential hydrocarbon bearing zones. 
A further composite parameter which can be expressed by the generalized 
relationship (9a) is attenuation, .alpha., i.e. the relative attenuation 
(typically corrected for temperature and spreading loss) measured by the 
microwave electromagnetic propagation tool ("EMP"-46 of FIG. 1). The 
relationship for this parameter is set forth above (9b), and will be 
considered momentarily. First, and consistent with the teachings of my 
U.S. Pat. No. 4,092,583, consider that the measured attenuation of the 
bulk formation (designated .alpha.) can be expressed as 
EQU .alpha.=.phi..sub.w .alpha..sub.wc +(1-.phi..sub.w).alpha..sub.m (33) 
where .alpha..sub.wc is the attenuation attributable to the formation water 
(i.e., its composite water, in accordance with the teachings hereof) and 
.alpha..sub.m is the attenuation attributable to the formation matrix. 
Since .alpha..sub.m is very small compared to .alpha..sub.w, one can write 
EQU .alpha.=.phi..sub.w .alpha..sub.wc (34) 
This relationship expresses that the bulk formation attenuation is 
volumetrically "adjusted" by a factor .phi..sub.w to take account of the 
fact that loss is essentially only occurring in that fraction of the bulk 
formation occupied by the water. Returning, now, to relationship (9b), we 
have 
##EQU21## 
where .alpha..sub.wf is the attenuation attributable to the free water 
(i.e. the attenuation which one would measure with the "EMP" logging 
device in a theoretical environment consisting exclusively of the 
formation free water), .alpha..sub.wb is the attenuation attributable to 
the bound water (i.e. the attenuation which one would measure with the 
"EMP" logging device in a theoretical environment consisting exclusively 
of the formation bound water), and .alpha..sub.wc is the attenuation 
attributable to the composite water (i.e. the attenuation which one would 
measure with a "EMP" logging device in a theoretical environment 
consisting exclusively of the actual formation water). 
Solving relationship (9b) for the bound water fraction, S.sub.wb /S.sub.w, 
yields the relationship (9c) first set forth above: 
##EQU22## 
In the form of the present invention, .alpha..sub.wf and .alpha..sub.wb (or 
these parameters multiplied by water filled porosity, .phi..sub.w, to 
obtain "bulk" variables .phi..alpha..sub.wf and .phi..alpha..sub.wb) are 
determined using attenuation and travel time (or velocity) measurements 
taken with an electromagnetic propagation logging device such as "EMP" 46 
of FIG. 1. The conductivity (generally of the formation invaded zone) 
obtained using the "EMP" device, designated .sigma..sub.EMP, can be 
expressed as 
EQU .sigma..sub.EMP =.alpha.t.sub.pl /K (35) 
where K is a constant, t.sub.pl is the measured travel time through the 
formations, and .alpha. is the bulk attenuation determined from the 
measured attenuation corrected for spreading loss and temperature, .alpha. 
being .phi..sub.w .alpha..sub.wc (relationship (34) above). While the 
relationship (35) for conductivity is expected to hold substantially 
independent of the salinity of the formation water, it has been observed 
that frequently .sigma..sub.EMP exceeds the conductivity measured from 
other tools. An explanation for the observed differences in conductivity 
is that not all of the losses represented by the bulk attenuation 
measurement .alpha. are due to the conductivity or salinity of the 
formation water. Extraordinary losses are believed to occur in the 
presence of bound water, these losses being more dielectric than 
conductive in nature. Applicant has discovered that treating bound water 
losses separate from the ordinary expected free water losses resolves the 
problem and produces more realistic values of .sigma..sub.EMP. In 
accordance with a feature of the invention, and as will be described, an 
attenuation representative variable is determined that is, inter alia, 
more appropriate for use in obtaining .sigma..sub.EMP. In the example 
below, this attenuation representative variable is the free water variable 
.phi..sub.w .alpha..sub.wf. The determined variable is also useful in 
conjunction with other techniques where attenuation is utilized as an 
input or a correction. 
Referring to FIG. 12, there is shown implementation of the computing module 
510 of FIG. 1 which is utilized to generate a signal representative of the 
bound water fraction, S.sub.wb /S.sub.w. A pair of difference circuits 501 
and 502 are provided. The positive input terminal of circuit 501 receives 
a signal representative of the quantity .alpha..sub.wc and the negative 
input terminal of circuit 501 receives a signal representative of the 
quantity .alpha..sub.wf. The positive input terminal of circuit 502 
receives a signal representative of the quantity .alpha..sub.wb, and the 
negative input terminal of circuit 502 receives the signal representative 
of the quantity .alpha..sub.wf. The outputs of difference circuits 501 and 
502 are respectively coupled to a ratio circuit 503 which produces a 
signal proportional to the ratio of the output of circuit 501 divided by 
the output of circuit 502. The output of ratio circuit 503 is accordingly 
a signal representative of the bound water fraction, S.sub.wb /S.sub.w, in 
accordance with relationship (9c). In actuality, and as will be clarified 
shortly, the inputs to computing module 510 may each have a common 
multiplier, .phi..sub.w. 
The manner in which the inputs to computing module 510 can be developed 
will now be described. In particular, one preferred technique for deriving 
values of .alpha..sub.wf and .alpha..sub.wb (or, of related bulk 
attenuation variables .phi..sub.w .alpha..sub.wf and .phi..sub.w 
.alpha..sub.wb) is as follows: Log values of .alpha. (attenuation) and 
t.sub.pl (travel time) are initially obtained over a range of depth levels 
of interest (e.g., using EMP device 46 of FIG. 1--these outputs being 
indicated as being available from processing circuitry 51). The obtained 
values of .alpha. and t.sub.pl are cross plotted, as shown in the 
frequency cross plot of FIG. 13. The values of .alpha. may first be 
corrected for temperature and for spreading loss. The cross plot of FIG. 
13 can be initially understood by recognizing that higher porosity 
generally results in higher values of both attenuation and travel time (at 
least, when that porosity contains water). This is because the water is 
much lossier than the rock matrix (thus: greater attenuation) and the 
velocity of the electromagnetic energy through water is lower than through 
the matrix (thus: greater travel time). Accordingly, increasing values of 
t.sub.pl and .alpha. on the cross plot generally correspond to increasing 
values of porosity. It can be noted that .alpha. could alternatively be 
cross-plotted against other non-conductivity related measurements 
reflecting total porosity, .phi..sub.t, such as .phi..sub.ND, previously 
described. 
The point designated t.sub.pm on the t.sub.pl axis represents the travel 
time through the formation matrix. Two trend lines, designated as the 
"free water trend line" and the "bound water trend line" are constructed 
by starting at the point t.sub.pm and drawing lines through the 
approximate bottom and top edges of the main cluster of points on the 
cross plot. These trend lines can be understood in the following terms: In 
those portions of the formations containing substantially only free water, 
both t.sub.pl and .alpha. will increase with porosity, with the increase 
in travel time being dependent upon the volume of water and the increase 
in attenuation being dependent upon both the volume of water and its 
conductivity. Accordingly, the slope of the free water trend line will 
depend upon the conductivity or lossiness associated with the free water. 
The same will generally be true of those portions of the formations in 
which substantially all of the water is bound water. However, in this 
case, attenuation will be a function of not only the volume of water and 
its conductivity, but also of the generally higher losses, included 
dipolar losses, associated with the bound water. Accordingly, the bound 
water trend line usually has substantially greater slope than the free 
water trend line. It will be understood that these trends representing the 
relationships between attenuation and travel time in a substantially free 
water region (such as a clean sand) and a bound water region (such as a 
shale) could be determined initially from logs taken in such formation 
regions. Also, it will be understood that these relationships are 
determinable functions which need not necessarily be linear, but are 
illustrated as being linear in the graph of FIG. 13. 
Having established free water and bound water trend lines (or functions), 
one can now, at each depth level of interest, obtain a free water 
attenuation quantity representative of the attenuation attributable to the 
formations (surrounding the depth level of interest) as if substantially 
all of the water in the formations was free water. Similarly, one can 
derive a bound water attenuation quantity representative of the 
attenuation attributable to said formations (surrounding the depth level 
of interest) as if substantially all of the water in the formations was 
bound water. Using these quantities, in conjunction with the measured 
attenuation at the depth level of interest, one can then determine the 
bound water fraction in the formations surrounding the particular depth 
level. With reference to FIG. 13, consider the illustrated individual 
point (.alpha., t.sub.pl) and the vertical line drawn therethrough. At the 
particular measured value of t.sub.pl, the intersection with free water 
trend line indicates the attenuation value that one would have measured if 
the water in the pore spaces of this particular formation contained 
exclusively free water (i.e., .phi..sub.w .alpha..sub.wf) whereas the 
intersection with the bound water trend line indicates the attenuation 
that would have been measured if the pore spaces of this formation 
contained exclusively bound water (i.e., .phi..sub.w .alpha..sub.wb). In 
actuality, the measured attenuation (.alpha.=.phi..sub.w .alpha..sub.wc) 
is an attenuation which has a value between these two extreme values, and 
the total water in the pore spaces can be considered as a composite water 
having attenuation .alpha..sub.wc. Accordingly, it is seen that 
relationship (9c) and the output of computing module 510 represents a 
linear apportionment between the two extreme values and yields the bound 
water fraction, S.sub.wb /S.sub.w. (Note that the multiplier .phi..sub.w 
before each term will be cancelled in the output of computing module 510 
if .phi..sub.w .alpha..sub.wc, .phi..sub.w .alpha..sub.wf and .phi..sub.w 
.alpha..sub.wb are used as the input quantities.) 
In addition to the use of .phi..sub.w .alpha..sub.wf and .phi..sub.w 
.alpha..sub.wb in obtaining the bound water fraction, the bulk formation 
attenuation if all the water was free water (i.e., .phi..sub.w 
.alpha..sub.wf) is useful, as first noted above, in determining 
.sigma..sub.EMP, since attenuation due to whatever bound water is present 
will not then result in an unduly high value of .sigma..sub.EMP. In 
particular, .sigma..sub.EMP can be determined from 
##EQU23## 
which is a modified form of relationship (35) wherein the bulk free water 
attenuation (.phi..sub.w .alpha..sub.wf) is substituted for the bulk 
composite water attenuation (.phi..sub.w .alpha..sub.wc which is the 
equivalent of the measured .alpha. in accordance with (34) above). 
An alternative technique for obtaining the bulk free water attenuation, 
.phi..sub.w .alpha..sub.wf, is to use the apparatus of FIG. 14. A ratio 
circuit 431 receives at its inputs signals representative of .alpha. and 
.phi..sub.w, both as determined from measurements taken with an EMP device 
46 (FIG. 1) in a clean non-hydrocarbon-bearing region of the formations in 
which substantially all of the water present is free water. (The signal 
representative of .phi..sub.w may be obtained, for example, using the 
technique of my U.S. Pat. No. 4,092,583.) The ratio .alpha./.phi..sub.w, 
in this region, will be representative of .alpha..sub.wf in accordance 
with relationships (34) and (9b), where S.sub.wb =0 for this case. In 
particular 
##EQU24## 
EQU .alpha.=.phi..sub.w .alpha..sub.wf (when S.sub.wb =0) (38) 
so that .alpha..sub.wf =.alpha./.phi..sub.w when S.sub.wb =0. Having 
obtained the parameter .alpha..sub.wf for the formations, the variable 
.phi..sub.w .alpha..sub.wf (i.e., the bulk free water attenuation) can now 
be determined at a particular depth level of interest by multiplying the 
output of ratio circuit 431 by a signal representative of .phi..sub.w at 
that depth level; this being implemented by multiplier circuit 432. A 
further multiplier circuit 433 can then be employed to obtain a signal 
representative of .sigma..sub.EMP in accordance with relationship (36). It 
will be understood that analagous circuitry could be used to obtain a 
corresponding bound water parameter, .alpha..sub.wb, from information in a 
shaley region, and then the bulk bound water attenuation at specific depth 
levels of interest could be obtained using a multiplier circuit to produce 
a signal representative of .phi..sub.w .alpha..sub.wb. The signals 
representative of .phi..sub.w .alpha..sub.wf and .sigma..sub.EMP can also 
be recorded, if desired, by recorder 90 of FIG. 1. 
It can be noted, in the context of obtaining either the bound water or free 
water related values, that non-linear interpolation can be employed, if 
desired (e.g., in FIG. 13). Further, since t.sub.pl may be affected by 
residual hydrocarbons left in the formation near the borehole, the 
indicated attenuation corresponding to free or bound water conditions may 
be slightly inaccurate. However, since both t.sub.pl and .alpha. will 
decrease due to hydrocarbon effects, there is some compensation in the 
indicated bound or free water saturations. When .alpha..sub.wf or 
.phi..sub.w .alpha..sub.wf is determined, the hydrocarbon effects will 
lower corresponding t.sub.pl values and will produce slightly lower 
.alpha..sub.wf values and hence, when applied in conductivity 
measurements, lower .sigma..sub.EMP values. Use of a .phi..sub.t 
measurement (relatively independent of hydrocarbon effects) in place of 
t.sub.pl, in the technique illustrated in FIG. 13, may be advisable in 
some instances. 
The invention has been described with reference to particular embodiments, 
but variations within the spirit and scope of the invention will occur to 
those skilled in the art. For example, while circuitry has been described 
for generating analog signals representative of the desired quantities, it 
will be understood that a general purpose digital computer could readily 
be programmed to implement the techniques as set forth herein. Also, while 
conductivity values have been utilized for purposes of illustration, it 
will be recognized that the inverses of values utilized herein could be 
employed in conjunction with the inverse of conductivity; i.e., 
resistivity.