Method of correcting formation resistivity well logs for the effects of formation layer inclination with respect to the wellbore

A method for correcting response of an induction logging instrument for inclination of earth formations with respect to an axis of the instrument. The instrument has a transmitter and a plurality of receivers at spaced apart locations. The method includes calculating expected receiver responses of simulated media having different conductivities. The calculations are performed for a plurality of different inclinations. The calculations are also performed for media having a plurality of different conductivity contrasts. 2-dimensional filters corresponding to a charge effect portion of each of the expected responses are calculated. 2-dimensional filters corresponding to a volumetric effect portion of each of the expected responses are calculated. An angle of inclination of the earth formations with respect to the instrument is determined. An approximate conductivity contrast of the earth formations is determined. Coefficients are interpolated between the 2-dimensional charge effect filters having simulated inclinations and contrasts closest to the angle of inclination and conductivity contrast of the formations. The interpolated filter coefficients are applied to measured receiver responses, generating charge effect-filtered measured responses. Coefficients are interpolated between the 2-dimensional volumetric effect filters having simulated inclinations and conductivity contrasts closest to the angle of inclination and conductivity contrast of the formations. The interpolated coefficients are applied to the charge effect-filtered measured responses to calculate corrected receiver responses.

BACKGROUND OF THE INVENTION 
1. Field of the Invention 
The present invention is related to the field of electrical resistivity 
logging of earth formations penetrated by wellbores. More specifically, 
the present invention is related to methods for correcting electrical 
resistivity well logs for the effects of apparent formation inclination on 
these logs. 
2. Description of the Related Art 
Formation resistivity well logs are commonly used to map subsurface 
geologic structures and to infer the fluid content within pore spaces of 
earth formations. Formation resistivity well logs include electromagnetic 
induction logs. Electromagnetic induction logs typically are measured by 
an instrument which includes a transmitter through which a source of 
alternating current (AC) is conducted, and includes receivers positioned 
at spaced apart locations from the transmitter. The AC passing through the 
transmitter induces alternating electromagnetic fields in the earth 
formations surrounding the instrument. The alternating electromagnetic 
fields induce eddy currents within the earth formations. The eddy currents 
tend to flow in "ground loops", which are most commonly coaxial with the 
instrument. The magnitude of the eddy currents can be related to the 
electrical conductivity (the inverse of the resistivity) of the earth 
formations. The eddy currents, in turn, induce voltages in the receivers 
which, generally speaking, are proportional to the magnitude of the eddy 
currents. Various circuits are provided in the instrument to measure the 
induced voltages, and thus determine the conductivity (and the 
resistivity) of the earth formations. 
One simplifying assumption which is made in relating the receiver voltage 
measurements to the conductivity of the earth formations is that the 
ground loops are positioned entirely within a portion of the earth 
formation having substantially circumferentially uniform conductivity. 
This assumption fails in cases where layers of the earth formations are 
not perpendicular to, but are inclined with respect to, the axis of the 
wellbore (and consequently the axis of the instrument). A boundary 
separates two layers which can have different conductivities. As the 
instrument traverses the wellbore so as to approach one of these 
boundaries, a portion of one of the ground loops can be located within one 
layer of the earth formation, and another portion of the same ground loop 
can be positioned in the other layer having an entirely different 
conductivity. The eddy current flowing in this ground loop, and 
consequently the voltage induced in the receivers, will be affected by 
this occurrence. 
A particularly difficult problem arises when the axis of the instrument is 
not perpendicular to, but is inclined with respect to the formation 
layers. If the layers are not perpendicular to the axis of the instrument, 
the conductivity of the media surrounding the instrument can vary 
circumferentially, causing the inferences about the conductivity from the 
measurements of induced voltage to be in error. As a practical matter, the 
effect on the induction resistivity log of inclined formation layers, is 
the same either when the layers are inclined from horizontal and are 
penetrated by a vertical wellbore, or when the wellbore itself is inclined 
from vertical and penetrates substantially horizontal formation layers in 
the earth formations. What is important in causing the effect of inclined 
layers is the relative inclination of the layers with respect to the axis 
of the wellbore, and correspondingly, the axis of the instrument. 
Various investigators have attempted to quantify the effect on induction 
logs of inclined formation layers so that induction logs run under such 
conditions could be recalculated to represent the measurements that would 
have been made if the layers were at right angles with respect to the 
instrument axis. In thus recalculating the well logs, better indications 
of the actual conductivity (resistivity) of the earth formations could be 
obtained. For example, in "Theory of Induction Sonde in Dipping Beds", R. 
Hardman et al, Geophysics, vol. 51, no. 3, pp. 800-809, Society of 
Exploration Geophysicists (1986), a forward model is described which 
provides simulated responses of induction logging sondes within media 
having preselected conductivity, thickness and formation layer inclination 
with respect to the simulated instrument. Such forward modelling has been 
improved upon, but in the end still retains a fundamental drawback common 
to such modelling when used for determining true resistivity from measured 
data. The simulated values must be compared to the measured values, the 
difference between them calculated, the model updated, and the process 
repeated until convergence is achieved between the model and the measured 
data. This process can be time consuming and difficult to perform, 
particularly at the wellsite. 
An improved method of correcting induction logs for formation layer 
inclination is described in U.S. Pat. No. 5,184,079 issued to Barber. The 
method described in the Barber '079 patent includes convolving the 
apparent resistivity data with an inverse filter to generate corrected 
resistivity data. The inverse filters are generated by calculating log 
response functions for models of inclined formation layers for various 
earth formations. A drawback to the method described in the Barber '079 
patent is that the inverse filters are one-dimensional. The 
one-dimensional filters only correct the response of the induction log for 
the increased effect of the formation layers axially located above and 
below the instrument (the so-called "shoulder bed" effect) when the layers 
are inclined with respect to the instrument. The method disclosed by 
Barber does not account for the effect of charge buildup which occurs at 
the boundaries of the layers of the earth formations. This so-called 
"charge effect" makes up a substantial portion of the total response of 
induction logs where the boundaries are inclined with respect to the axis 
of the instrument. 
A more complete correction for the response of induction logs for inclined 
formations layers should take into account the charge effect and a 
correction for a so-called "volumetric effect". The volumetric effect is 
partially related to an increase in the shoulder bed effect caused by the 
relative inclination of the formation layers with respect to the 
instrument, but the volumetric effect cannot be completely corrected with 
a one-dimensional filter. 
Accordingly it is an object of the present invention to provide a 
correction method for induction resistivity well logs for formation layer 
inclination which provides improved performance over the methods of the 
prior art. 
It is another object of the present invention to provide a correction 
method for induction resistivity well logs for formation layer inclination 
which quantifies both the charge effect and the volumetric effect on 
induction logs subject to crossing inclined formation layers. 
SUMMARY OF THE INVENTION 
The present invention is a method of correcting the response of an 
induction well logging instrument for the effects of inclination of earth 
formations with respect to an axis of the instrument. The induction 
logging instrument has a transmitter and a plurality of receivers at 
axially spaced apart locations. The method includes calculating expected 
responses of the receivers on the instrument in simulated media having 
different conductivities. The step of calculating includes performing the 
calculation at a plurality of different inclinations of the media with 
respect to the axis of the instrument. The step of calculating also 
includes performing the calculation for media having a plurality of 
conductivity contrasts. The method includes calculating 2-dimensional 
filters corresponding to a charge effect portion of each one of the 
simulated responses and calculating 2-dimensional filters corresponding to 
a volumetric effect portion of each one of the simulated responses. An 
angle of inclination of the layers of the earth formations with respect to 
the axis of the instrument is determined. An approximate conductivity 
contrast of the earth formations is determined. Coefficients are 
interpolated between ones of the 2-dimensional charge effect filters 
having simulated inclinations and conductivity contrasts closest to the 
angle of inclination and conductivity contrast of the earth formations. 
The interpolated charge effect filter coefficients are applied to measured 
responses of the instrument. Coefficients are interpolated between ones of 
the 2-dimensional volumetric effect filters having simulated inclinations 
and conductivity contrasts closest to the angle of inclination and 
conductivity contrast of the earth formations, these coefficient are 
applied to the charge effect filtered measured responses of the instrument 
to generate corrected responses.

DESCRIPTION OF THE PREFERRED EMBODIMENT 
FIG. 1 shows an induction well logging instrument 10 disposed in a wellbore 
2 drilled through earth formations. The earth formations are shown 
generally at 6, 7, 8, 9, 12, 13 and 14. The instrument 10 is typically 
lowered into the wellbore 2 at one end of an armored electrical cable 22 
by means of a winch 28 or similar device known in the art. An induction 
well logging instrument which will generate appropriate signals for 
performing the process of the present invention is described, for example, 
in U.S. Pat. No. 5,452,761 issued to Beard et al. The instrument described 
in the Beard et al '761 patent is meant only to serve as an example of, 
and is not meant to be an exclusive representation of induction well 
logging instruments which can generate signals usable for performing the 
process of the present invention. 
The instrument described in the Beard et al '761 patent is therefore not to 
be construed as a limitation on the present invention. The instrument 
described in Beard et al '761 patent, however, has certain advantages as 
concerns the method of the present invention which will be further 
explained. 
The instrument 10 can include a telemetry/signal processing unit 20 (SPU). 
The SPU 20 can include a source of alternating current (not shown 
separately). The alternating current is generally conducted through a 
transmitter 16 disposed on the instrument 10. Receivers 18A-18F can be 
disposed at axially spaced apart locations along the instrument 10. The 
SPU 20 can include receiver circuits (not shown separately) connected to 
the receivers 18A-18F for detecting voltages induced in each of the 
receivers 18A-18F. The SPU 20 can also impart signals to the cable 22 
corresponding to the magnitude of the voltages induced in each of the 
receivers 18A-18F. It is to be understood that the number of transmitters 
and receivers, and the relative geometry of the transmitter 16 and the 
receivers 18A-18F shown in the instrument in FIG. 1 is not meant to be a 
limitation on the present invention. 
As is understood by those skilled in the art, the alternating current 
passing through the transmitter 16 induces eddy currents in the earth 
formations 6, 7, 8, 9, 12, 13, 14. The eddy currents correspond in 
magnitude both to the electrical conductivity of the earth formations 6, 
7, 8, 9, 12, 13, 14 and to the relative position of the particular earth 
formation with respect to the transmitter 16. The eddy currents in turn 
induce voltages in the receivers 18A-18F, the magnitude of which depends 
on both the eddy current magnitude and the relative position of the earth 
formation with respect to the individual receiver 18A-18F. 
The signals, corresponding to the voltages induced in each receiver 
18A-18F, can be transmitted along the cable 22 to surface electronics 24. 
The surface electronics 24 can include detectors (not shown separately) 
for interpreting the signals transmitted from the instrument 10, and a 
computer 26 to perform the process according to the present invention on 
the signals transmitted thereto. It is to be understood that the SPU 20 
could also be programmed to perform the process of the present invention. 
Processing the receiver signals in the computer 26 is a matter of 
convenience for the system designer and is not to be construed as a 
limitation on the present invention. 
The voltages induced in each receiver 18A-18F correspond to apparent 
electrical conductivity of all of the media surrounding the instrument 10. 
The media comprise the earth formations 6, 7, 8, 9, 12, 13, 14 and the 
drilling mud 4 in the wellbore 2. The degree of correspondence between the 
voltages induced in a particular receiver, and the electrical conductivity 
of the particular earth formation axially disposed between the particular 
receiver and the transmitter 16, can depend on the relative inclination of 
the layers of the earth formations, such as formation 12, and the axis of 
the instrument 10. 
Some of the earth formations, such as the one shown at 6, can be 
substantially horizontal, but perpendicular to the axis of the wellbore 2 
when the wellbore 2 is substantially vertical. Other formations, such as 
the ones shown at 8 and 9, can be geologically inclined, and inclined with 
respect to the axis of the wellbore 2 when the wellbore 2 is substantially 
vertical. Other earth formations, such as 12, can be only slightly 
geologically inclined, but have high inclination with respect to the axis 
of the wellbore 2 because the wellbore 2 is itself highly inclined from 
vertical. It generally does not matter for the purposes of the present 
invention whether the earth formation is geologically inclined, such as 
formation 8 and is penetrated by a portion of the wellbore 2 which is 
substantially vertical, or whether the earth formation is substantially 
horizontal, such as formation 14 and is penetrated by a portion of the 
wellbore 2 which is inclined from vertical. For purposes of the present 
invention, the axis of the instrument 10 is assumed to be substantially 
coaxial with the wellbore 2. The significance of the formation inclination 
with respect to the instrument 10 on the instrument 10 response will be 
further explained. 
A particular advantage of the induction logging apparatus disclosed in the 
Beard et al '761 patent is that it includes six receivers (such as 18A-18F 
in FIG. 1) each at a different axial spacing from the transmitter. It is 
to be understood that the number of receivers is not to be construed as a 
limitation on the invention. A larger number of axially spaced apart 
receivers can provide improved performance of the process of the present 
invention, as will be further explained, but the number of receivers may 
be practically limited by the design considerations of an instrument which 
would be commercially suitable for use in the wellbore 2. 
An apparatus suitable for generating measurements for the present invention 
having been explained, the theory of operation of the present invention 
will now be explained in more detail. The electric field vector, 
represented by E, generated by the transmitter 16 in the induction logging 
instrument 10 satisfies the following differential equation, wherein the 
time factor of the current in the transmitter 16 is represented by 
e.sup.-i.omega.t, and .omega. represents the angular frequency of the 
current in the transmitter 16: 
EQU .gradient..times..gradient..times.E(r)=K.sup.2 (r)E(r)+i.omega..mu.J(r) (1) 
In equation (1) r represents the coordinates of an observation point, J(r) 
represents the electric current density at r, .mu. represents the magnetic 
permeability of the medium (which can include earth formations such as 6, 
7, 8, 9, 12, 13, 14) surrounding the instrument 10, and K(r) represents 
the wavenumber of the medium surrounding the instrument 10 at the 
observation point r. The wavenumber can be expressed by: 
##EQU1## 
If the magnetic permeability and the dielectric permittivity (represented 
by .epsilon. in equation (2)) are assumed to be constant (which assumption 
is reasonable for most earth formations, particularly when energized with 
alternating current at induction logging frequencies of about 10-150 KHz), 
the wavenumber substantially represents the conductivity of the medium 
surrounding the instrument 10. Introducing a notation of a "background" 
wavenumber (or conductivity), equation (1) can be rewritten in the form: 
EQU .gradient..times..gradient..times.E(r)-K.sub.b.sup.2 
E(r)=i.omega..mu.J(r)+K.sup.2 (r)-K.sub.b.sup.2 !E(r) (3) 
wherein K.sub.b represents the wavenumber of the background. By 
incorporating Green's function g.sub.0 (r, r"), the solution of equation 
(3) can be shown by the following expression: 
##EQU2## 
In equation (4), the first term on the right hand side E.sub.0 (r), 
referred to as the incident electric field, can be represented by the 
following expression: 
EQU E.sub.0 (r)=i.omega..mu..intg.dr'G.sub.0 (r,r')J(r') (5) 
wherein G.sub.0 (r,r') represents the dyadic Green's function. The second 
term (the first integral with respect to r") on the right hand side of 
equation (4), as will be further explained, is referred to as the 
"volumetric" term. The third term (the second integral with respect to r") 
on the right hand side of equation (4), as will be further explained, is 
referred to as the "charge" term. 
The magnitude of the charge term, as will be further explained, results 
from buildup of electric charges at the boundary of two layers having 
different conductivities, through which eddy currents flow when the 
boundaries are not perpendicular to the axis of the instrument 10. As a 
practical matter, the instrument 10 is generally substantially coaxial 
with the wellbore 2, and previous references to the inclination of the 
earth formations with respect to the axis of the wellbore apply to this 
description as being with respect to the axis of the instrument 10. 
The charge term is substantially equal to zero when the axis of the 
instrument 10 is substantially perpendicular to the formation layers, 
because under this condition the electric field vector is typically 
perpendicular to the directional derivative of the wave number. The charge 
term, however, can be non-zero when the formation layers are not 
perpendicular to the axis of the instrument 10. The charge term can be 
referred to as the "charge-effect" term when the formation layers are 
inclined with respect to the axis of the instrument 10. 
The volumetric term can be different when the instrument 10 axis is 
inclined with respect to the formation layers than when the instrument 10 
axis is perpendicular to these layers. The difference between the 
volumetric term when the formation layers are perpendicular to the 
instrument 10 axis, and the volumetric term when the layers are inclined 
thereto is referred to as the volumetric effect. 
As is understood by those skilled in the art, the eddy currents induced in 
the media surrounding the instrument flow in generally circular ground 
loops substantially perpendicular to the axis of the instrument 10. When 
the instrument 10 axis is perpendicular to the earth formations (or layers 
of media having different electrical conductivities) the ground loops tend 
to flow within media having circumferentially uniform conductivity. This 
is not the case when the formations are inclined with respect to the 
instrument 10 axis. The different formations (media) actually traversed by 
a particular ground loop can be better understood by referring to FIGS. 
2A, 2B, 3A and 3B. FIG. 2A shows the instrument 10 traversing earth 
formations (media) having conductivities represented by .sigma..sub.1, 
.sigma..sub.2 and .sigma..sub.3. The formations in FIG. 2A are 
substantially perpendicular to the axis of the instrument 10. A ground 
loop 11 corresponding to one of the receivers (18A-18F in FIG. 1) is shown 
as surrounding the instrument but generally flowing within the formation 
represented by .sigma..sub.2. A linear perspective of the formation 
through which the ground loop 11 flows can be observed in FIG. 2B. 
FIG. 3A shows the instrument disposed within the same three formation 
layers, but in FIG. 3A the formations are inclined with respect to the 
axis of the instrument 10. As can be observed in FIG. 3B, the ground loop 
11 flows within all three different formations. A linear perspective of 
the ground loop 11 in FIG. 3A can be observed in FIG. 3B. Eddy currents 
flowing in ground loop 11 depend on the conductivities of the media 
.sigma..sub.1, .sigma..sub.2 and .sigma..sub.3 and on electric charge 
buildup at the boundaries of the media through which the ground loop 11 
passes. The amount of charge buildup at each boundary depends on the 
conductivity contrast between the media on each side of each boundary. 
The present invention provides a method for substantially filtering out the 
charge effect and then determining the magnitude of the volumetric effect 
of formation layers which are inclined with respect to the axis of the 
instrument (10 in FIG. 1). The first part of the method includes 
determining a set of so-called "response functions" for each receiver 
(such as 18A-18F in FIG. 1) on the instrument (10 in FIG. 1). The term 
"response function" is more precisely defined as the actual response of 
each receiver on the instrument to an infinitely thin bed having infinite 
conductivity (where the product of conductivity and thickness of this bed 
is unity) interposed within an infinitely thick medium having zero 
conductivity. This is more commonly known as the "impulse response" of 
each receiver on the instrument. As a practical matter, however, "impulse" 
(zero-thickness) earth formations do not exist. It is therefore preferable 
to simulate the instrument response by designing an appropriate set of 
"step functions", each of which represents known values of conductivity 
contrast across a discrete boundary. 
Besides being affected by the angle of inclination, the magnitudes of the 
charge effect and the volumetric effect are affected primarily by the 
ratio of conductivity (referred to herein as the "conductivity contrast") 
across the boundary, rather than the absolute magnitudes of the 
conductivities. Step function responses can be calculated for a number of 
predetermined inclination angles, and also for a number of different 
conductivity contrast values at each of these inclination angles. 
The first derivative of the instrument response to the step functions can 
then be calculated. The normalized first derivative represents, in fact, 
the impulse response of the instrument. For example, a step function of 
conductivity can be exemplified by formation conductivities which satisfy 
the following relationship: 
##EQU3## 
where a and b are arbitrary positive constants representing conductivity 
values on either side of the boundary and z represents the axial position 
of the instrument within simulated layers of the media surrounding the 
instrument. z is usually set to zero at the position of the boundary 
itself. Using a linear approximation known in the art as the "Doll 
approximation" for the response of the induction logging instrument (10 in 
FIG. 1), the instrument 10 response can be expressed as a convolution. 
More specifically, the instrument 10 response can be represented by the 
following expression: 
##EQU4## 
where R represents the response function of the instrument 10. By 
substitution of the relationship of equation (6) into the expression of 
equation (7), an expression for determining the conductivities in terms of 
the response functions can be shown as: 
##EQU5## 
The first derivative of the expression in equation (8) provides a solution 
for the response function R, shown in the following expression as: 
##EQU6## 
The charge term and the volumetric term of the total response of the 
instrument are additive, as shown in equation (4). Therefore, the response 
functions of the instrument 10 to an up-going step function, R.sup.u (z), 
(meaning that the conductivity increases across the boundary) and to 
down-going step function, R.sup.d (z), (meaning that the conductivity 
decreases across the boundary) can be expressed as the sums of the 
volumetric responses and the charge-effect responses according to the 
following expressions, first for the up-going response: 
EQU R.sup.u (z)=R.sub.vol (z)+R.sup.u.sub.chg (z) (10) 
and then for the down-going response: 
EQU R.sup.d (z)=R.sub.vol (z)+R.sup.d.sub.chg (z) (11) 
where R.sub.vol (z) represents the volumetric response, and R.sup.d.sub.chg 
(z) and R.sup.u.sub.chg (z), respectively, represent the up-going and 
down-going charge-effect responses. 
As can be determined from the expression in equation (4), and has been 
confirmed in the process of generating response models, the up-going and 
down-going charge-effect responses are substantially equal in magnitude 
and opposite in sign. Averaging the up-going and down-going instrument 
responses can therefore provide a direct solution for the volumetric 
response of the instrument. The volumetric response can then be calculated 
by the following expression: 
##EQU7## 
The individual up-going and down-going charge-effect response functions 
can then be obtained by the following expressions: 
EQU R.sup.u.sub.chg (z)=R.sup.u (z)-R.sub.vol (z) (13) 
EQU R.sup.d.sub.chg (z)=R.sup.d (z)-R.sub.vol (z) (14) 
In the present invention, the response functions can be expressed in vector 
notation to represent a collection of the response functions attributable 
to each receiver (such as 18A-18F in FIG. 1) on the instrument (10 in FIG. 
1). For example: 
EQU R.sup.u =(R.sup.u.sub.1,R.sup.u.sub.2, . . . R.sup.u.sub.N), 
EQU R.sup.d =(R.sup.d.sub.1,R.sup.d.sub.2, . . . R.sup.d.sub.N), 
EQU R.sub.vol =(R.sub.vol1,R.sub.vol2, . . . R.sub.volN) (15) 
EQU R.sup.u.sub.chg =(R.sup.u.sub.chg1,R.sup.u.sub.chg2, . . . 
R.sup.u.sub.chgN) 
EQU R.sup.d.sub.chg =(R.sup.d.sub.chg1,R.sup.d.sub.chg2, . . . 
R.sup.d.sub.chgN) 
where 1, 2, . . . , N represents the ordinal location of the particular 
receiver (corresponding to 18A, 18B, . . . , 18F in FIG. 1) for which that 
component of the vector is determined. 
Response functions for the instrument of FIG. 1 can be observed by 
referring to FIGS. 4A and 4B. The graphs in FIGS. 4A and 4B show 
fractional response of the individual receivers as a function of the axial 
position of the instrument. The axial position of the boundary is located 
at zero on the coordinate axis. FIG. 4A represents the step function total 
response of the instrument of FIG. 1 to an up-going step function having a 
conductivity contrast of 100. The conductivity values used for the 
response function calculation in FIG. 4A can be observed by referring to 
the graph in FIG. 4C. 
FIG. 4B shows the down-going total response of the instrument. The 
conductivity values used in the response simulation of FIG. 4B can be 
observed by referring to FIG. 4D. 
The charge effect portion of the total response can be observed for the 
up-going and the down-going responses, respectively, in the graphs of 
FIGS. 5A and 5B. The graphs in FIGS. 5A and 5B show the charge effect 
response of each receiver in the instrument as a function of the axial 
position of the instrument. The boundary is located at zero on the 
coordinate axis. FIGS. 5A and 5B show graphically that the charge-effect 
response for each receiver is opposite in sign and the same amplitude when 
the conductivities of the media on either side of the boundary are 
reversed. 
The present invention determines, and subsequently substantially removes, 
the charge effect portion of the total response of the instrument by 
2-dimensional (2-D) filtering. Although the charge effect response 
function is somewhat different for each receiver the charge effect at any 
one of the receivers can be determined from the responses of the other 
receivers because all the receiver responses contain charge effects from 
the same boundary. For example: 
EQU R.sup.u.sub.chg =R.sup.u *F (16) 
EQU R.sup.d.sub.chg =R.sup.d *F (17) 
where F represents a collection of 2-D filters, and * represents 
convolution of the 2-D filter with the response function in the respective 
equations. 
Several constraints for the filter F can be inferred from the relationships 
expressed in equations (15), (16) and (17), namely: 
EQU R.sub.vol *F=0 (18) 
EQU R.sup.u.sub.chg *F=R.sup.u.sub.chg (19) 
By substitution: 
where F' can be determined by the expression: 
EQU R.sup.d.sub.chg *F=R.sup.d.sub.chg (20) 
EQU R.sub.vol =R.sup.u *F' (21) 
EQU R.sub.vol =R.sup.d *F' (22) 
EQU F'=.delta.-F, .delta.=(.delta..sub.1,.delta..sub.2, . . . ,.delta..sub.N) 
(23) 
where .delta. represents the 2-dimensional Dirac delta function. 
Determining F' provides for substantial elimination of the charge effect 
from the data. 
In the present invention, 2-D filters which will substantially remove the 
charge effect response can be designed by simulation of the response of 
the instrument for step functions of various conductivity contrast and 
inclination angle. In the present invention, 2-dimensional filters can be 
designed for simulations of the instrument response at 0, 10, 20, 30, 40, 
50, 60, 70 and 75 degrees. Above about 75 degrees inclination, the 
analysis of the charge effect and volumetric effect of the present 
invention becomes somewhat inaccurate. Conductivity contrast values of 10 
and 100 can be used for the filter design. The filter coefficients can be 
solved by an optimization routine such as one described, for example, in 
"Practical Optimization", P. E. Gill and W. Murray, Academic Press, 
London, 1981. The routine described in the Gill and Murray reference is a 
least squares minimization. Since the filters are subject to the 
constraints described in equations (18) (19) and (20), the process of 
calculating filter coefficients can be referred to as constrained least 
squares optimization. It is to be understood that other methods of 
optimizing the response of the 2-D filters can be used, and that least 
squares optimization should not be construed as a limitation on the 
invention. 
It is also to be explicitly understood that generating 2-D filters at 
predetermined inclination angles and conductivity contrasts, and storing 
such filters in a database for use in calculating corrected induction log 
responses is provided only as a convenience to the system designer. It is 
contemplated that response models, and their resultant 2-D filters, could 
be generated for any selected angle of inclination and any conductivity 
contrast values, so that a portion of a well log could be adjusted 
according to the present invention without the need to calculate the 2-D 
filters beforehand. It is also contemplated that filters could be 
calculated for the exact angle of inclination and conductivity contrast 
encountered on the well log for that portion of the well log being 
recalculated, so that interpolation between the filters at preselected 
inclination angles and conductivity contrasts is not necessary. 
The 2-D charge effect filters thus generated can be stored in a database 
for use in correcting the response of actual well log data. When actual 
log data from a wellbore are processed, the 2-D filters in the database 
can be adjusted by an adaptive algorithm to generate a 2-D filter for a 
particular portion of the well log data being processed. For example, 
inclination angle can be provided from external sources, such as dip 
measuring logs, geologic information, and directional surveys of the 
wellbore. The filter coefficients can be generated by interpolation 
between the coefficients of the charge effect filters stored in the 
database. The conductivity contrast can be selected by sampling a selected 
axial interval of the well log data, calculating first derivatives of the 
responses of each one of the receiver responses, and normalizing the 
calculated derivatives of the responses. Axial segments of receiver 
responses on either side of the boundaries can be averaged to estimate 
conductivity on both sides of each boundary. Generally the conductivity 
contrast selected from the database should be the higher value which most 
closely matches the contrast calculated for the selected axial interval. 
Examples of 2-D filters corresponding to receivers 18C of FIG. 1 and 18F of 
FIG. 1 are shown, respectively, in FIGS. 6A and 6B. After application of 
the 2-D charge effect correction filters, the receiver responses represent 
responses uncorrected only for the volumetric effect. The volumetric 
effect at an inclination angle of 60 degrees can be observed by referring 
to FIGS. 7A and 7B. In FIG. 7A, a simulated conductive layer having a 
thickness of 40 feet and conductivity of 1 S/m is embedded within a more 
resistive medium having conductivity of 0.01 S/m. Simulated responses for 
each of the receivers (18A-18F in FIG. 1) on the instrument are shown for 
an inclination angle of 60 degrees after application of the charge effect 
2-D filter. Simulated receiver responses for the same conductivity media 
at an inclination angle of zero are shown in FIG. 7B. While there is 
substantial agreement between the 60 degree and zero degree responses, a 
further adjustment to the 60 degree response is still warranted. The 
adjustment which is needed results from the volumetric effect. 
The magnitude of the volumetric response can be observed by referring to 
FIGS. 8A and 8B, which show, respectively, the volumetric responses at 
inclination angles of 60 and zero degrees for the step function used to 
generate the total responses as shown in FIGS. 4A and 4B. As previously 
explained, the volumetric response is not dependent on the direction of 
travel across the boundary, so there is no need to display up-going and 
down-going responses. The adjustment for the volumetric response can be 
provided by another step of 2-D filtering. A 2-D filter for the response 
of the instrument can be designed so as to make the response shown in FIG. 
8A across an inclined boundary substantially match the response of the 
instrument across a horizontal boundary as shown in FIG. 8B. 2-D filters 
can be generated and stored in the database corresponding to the same set 
of boundary inclinations and conductivity contrasts as described for the 
charge effect filters. The filter coefficients applied to actual well log 
data at inclinations and contrasts not precisely matching the database 
values can be generated by interpolation as previously described herein. 
The improvement in the calculated receiver response after application of 
the volumetric effect filter can be observed in FIGS. 9A and 9B. FIGS. 9A 
and 9B are graphs of simulations of receiver responses for the same 
conductivity media as shown for the graphs in FIGS. 7A and 7B, with FIG. 
9A showing volumetric effect corrected response at 60 degrees inclination, 
and FIG. 9B showing the same receiver responses at zero inclination. FIGS. 
9A and 9B show substantial agreement, indicating that the combination of 
2-D charge effect and 2-D volumetric effect filters substantially removes 
the effects of formation layer inclination from the response of the 
induction instrument. 
It is to be understood that 2-D filtering to remove the charge effect and 
2-D filtering to remove the volumetric effect can be performed in reverse 
order, or can be combined into a single step of filtering because the 
volumetric effect and the charge effect are additive, as described in 
equation (4). 
The description of the invention provided herein is meant only to serve as 
an example of a method of correcting induction logs for the effects of 
formation layer inclination. Those skilled in the art will readily devise 
other embodiments of the method which do not depart from the spirit of the 
invention. Accordingly, the invention should only be limited in scope by 
the attached claims.