U.S. patent application number 10/331597 was filed with the patent office on 2003-05-15 for 2-d inversion of multi-component induction logging data to resolve anisotropic resistivity structure.
This patent application is currently assigned to Baker Hughes, Inc.. Invention is credited to Mezzatesta, Alberto, Zhang, Zhiyi.
Application Number | 20030093223 10/331597 |
Document ID | / |
Family ID | 25172592 |
Filed Date | 2003-05-15 |
United States Patent
Application |
20030093223 |
Kind Code |
A1 |
Zhang, Zhiyi ; et
al. |
May 15, 2003 |
2-D inversion of multi-component induction logging data to resolve
anisotropic resistivity structure
Abstract
High Definition Induction Logging (HDIL) tools can provide
reliable information about the vertical and radial variations of
resistivity structure in isotropic media. The focusing technique
provides quantitative information about the resistivity variation
and qualitative information about invasion at the well site. This
type of logging tool utilizes transmitter-receiver arrays coaxial
with the borehole and thus cannot provide information about
anisotropy in vertical wells. This greatly limits the application
of array induction tools in the characterization of reservoirs with
finely laminated sand/shale sequences. A multi-component induction
tool, 3DEX.TM., has been developed by Baker Atlas and Royal Dutch
Shell. It provides the much needed ability to detect anisotropy for
sand-shale laminated reservoirs. Data from such a logging tool are
inverted to give an estimate of vertical and horizontal resistivity
in a vertical borehole. 3DEX.TM., however, lacks the radial
resolution provided by array induction tools. Thus 3DEX.TM.may
encounter difficulties in looking through an invaded zone and
detecting the anisotropy in the formations. Joint inversion of HDIL
and 3DEX.TM. data is able to identify parameters of the invaded
zone as well as of the anisotropic formations.
Inventors: |
Zhang, Zhiyi; (Houston,
TX) ; Mezzatesta, Alberto; (Houston, TX) |
Correspondence
Address: |
PAUL S MADAN
MADAN, MOSSMAN & SRIRAM, PC
2603 AUGUSTA, SUITE 700
HOUSTON
TX
77057-1130
US
|
Assignee: |
Baker Hughes, Inc.
|
Family ID: |
25172592 |
Appl. No.: |
10/331597 |
Filed: |
December 30, 2002 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
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10331597 |
Dec 30, 2002 |
|
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09798120 |
Mar 2, 2001 |
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6502036 |
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Current U.S.
Class: |
702/7 |
Current CPC
Class: |
G01V 3/28 20130101 |
Class at
Publication: |
702/7 |
International
Class: |
G01V 003/18 |
Claims
What is claimed is:
1. A method of logging of subsurface formations including a
plurality of layers each having a horizontal resistivity and a
vertical resistivity, the method comprising: (a) using a
multi-component electromagnetic logging tool in a borehole in the
subsurface formations for obtaining multi-component measurements
indicative of said resistivities of said layers, said borehole
associated with an invaded zone in said layers; (b) defining an
initial model of said plurality of layers, said initial model
including, for each of said plurality of layers, (i) a horizontal
resistivity, (ii) a vertical resistivity, (iii) a length of the
invaded zone, and (iv) a resistivity of said invaded zone; (c)
determining expected responses of the multi-component logging tool
to said model; (d) defining a data objective function related to a
difference between said expected responses and said measurements
made with the multi-component logging tool; (e) iteratively
updating said model thereby reducing a global objective function,
said global objective function comprising a sum of said data
objective function and a model objective function related to
changes in said model at each iteration.
2. The method of claim 1 wherein said electromagnetic logging tool
is conveyed on one of (i) a wireline, (ii) a drillstring, and (iii)
coiled tubing.
3. The method of claim 1 wherein said measurements made with said
multi-component logging tool measures comprise H.sub.xx, H.sub.zz
and H.sub.xz, measurements.
4. The method of claim 1 wherein said multi-component measures
further comprise H.sub.yy and H.sub.xy measurements.
5. The method of claim 1 wherein defining said initial model
further comprises using measurements from an array logging tool and
deriving the initial model from said array resistivity
measurements.
6. The method of claim 1 wherein determining said expected
responses further comprises using a forward modeling program.
7. The method of claim 1 wherein defining said global objective
function further comprises estimating a covariance of noise present
in said multi-component measurements.
8. The method of claim 1 wherein said model objective finction
includes a relative weight between resistivity components and
length components.
9. The method of claim 1 wherein iteratively updating the model
further comprises defining a sensitivity matrix relating the
observations to model parameters.
10. The method of claim 1 wherein iteratively updating said model
further comprises windowing of said measurements.
11. A method of logging of subsurface formations including a
plurality of layers each having a horizontal resistivity and a
vertical resistivity, the method comprising: (a) using a
multi-component logging tool in a borehole in the subsurface
formations for obtaining multi-component measurements indicative of
said resistivities of said layers, said borehole associated with an
invaded zone in said layers; (b) using an array logging tool in
said borehole for obtaining additional measurements primarily
indicative of said horizontal resistivity and properties of said
invaded zone; (c) defining an initial model of said plurality of
layers, said initial model including, for each of said plurality of
layers, (i) a horizontal resistivity, (ii) a vertical resistivity,
(iii) a length of the invaded zone, and (iv) a resistivity of said
invaded zone; (c) determining expected responses of the
multi-component logging tool and the array logging tool to said
model; (d) defining a data objective function related to a
difference between said expected responses and said measurements
made with the multi-component logging tool and the array logging
tool; (e) iteratively updating said model thereby reducing a global
objective function, said global objective finction comprising a sum
of said data objective function and a model objective function
related to changes in said model at each iteration.
12. The method of claim 1 wherein said multi-component logging tool
and said array logging tool are conveyed on one of (i) a wireline,
(ii) a drillstring, and (iii) coiled tubing.
13. The method of claim 11 wherein said measurements made with said
multi-component logging tool measures comprise H.sub.xx, H.sub.zz
and H.sub.xz measurements
14. The method of claim 12 wherein said measurements made with said
multi-component logging tool further comprise H.sub.yy, and
H.sub.xy measurements
15. The method of claim 11 wherein defining said initial model
further comprises using measurements from the array logging
tool.
16. The method of claim 11 wherein determining said expected
responses further comprises using a forward modeling program.
17. The method of claim 11 wherein defining said global objective
function further comprises estimating a covariance of noise present
in said multi-component and array logging measurements.
18. The method of claim 11 wherein said model objective function
includes a relative weight between resistivity components and
length components.
19. The method of claim 11 wherein iteratively updating the model
further comprises defining a sensitivity matrix relating the
observations to model parameters.
20. The method of claim 11 wherein iteratively updating said model
further comprises windowing of said measurements
Description
CROSS-REFERENCES TO RELATED APPLICATIONS
[0001] This application is related to U.S. patent application Ser.
No. 09/676,097 filed on Sep. 29, 2000.
BACKGROUND OF THE INVENTION
[0002] 1. Field of the Invention
[0003] The invention is related generally to the field of
interpretation of measurements made by well logging instruments for
the purpose of determining the properties of earth formations. More
specifically, the invention is related to methods for 2-D inversion
of induction logging data obtained with transverse induction
logging tools.
[0004] 2. Background of the Art
[0005] Electromagnetic induction and wave propagation logging tools
are commonly used for determination of electrical properties of
formations surrounding a borehole. These logging tools give
measurements of apparent resistivity (or conductivity) of the
formation that, when properly interpreted, are diagnostic of the
petrophysical properties of the formation and the fluids
therein.
[0006] The physical principles of electromagnetic induction
resistivity well logging are described, for example, in, H. G.
Doll, Introduction to Induction Logging and Application to Logging
of Wells Drilled with Oil Based Mud, Journal of Petroleum
Technology, vol. 1, p.148, Society of Petroleum Engineers,
Richardson Tex. (1949). Many improvements and modifications to
electromagnetic induction resistivity instruments have been devised
since publication of the Doll reference.. Examples of such
modifications and improvements can be found, for example, in U.S.
Pat. No. 4,837,517; U.S. Pat. No. 5,157,605 issued to Chandler et
al, and U.S. Pat. No. 5,452,761 issued to Beard et al.
[0007] U.S. Pat. No. 5,452,761 to Beard et al. having the same
assignee as the present application and the contents of which are
fully incorporated herein by reference, discloses an apparatus and
method for digitally processing signals received by an induction
logging tool having a transmitter and a plurality of receivers. An
oscillating signal is provided to the transmitter, which causes
eddy currents to flow in a surrounding formation. The magnitudes of
the eddy currents are proportional to the conductivity of the
formation. The eddy currents in turn induce voltages in the
receivers. The received voltages are digitized at a sampling rate
well above the maximum frequency of interest. The digitizing window
is synchronized to a cycle of the oscillating current signal.
Corresponding samples obtained in each cycle are cumulatively
summed over a large number of such cycles. The summed samples form
a stacked signal. Stacked signals generated for corresponding
receiver coils are transmitted to a computer for spectral analysis.
Transmitting the stacked signals and not all the individually
sampled signals, reduces the amount of data that needs to be stored
or transmitted. A Fourier analysis is performed of the stacked
signals to derive the amplitudes of in-phase and quadrature
components of the receiver voltages at the frequencies of interest.
From the component amplitudes, the conductivity of the formation
can be accurately derived.
[0008] A limitation to the electromagnetic induction resistivity
well logging instruments such as that discussed in Beard is that
they typically include transmitter coils and receiver coils wound
so that the magnetic moments of these coils are substantially
parallel only to the axis of the instrument. Eddy currents are
induced in the earth formations from the magnetic field generated
by the transmitter coil, and in the induction instruments known in
the art, these eddy currents tend to flow in ground loops which are
substantially perpendicular to the axis of the instrument. Voltages
are then induced in the receiver coils related to the magnitude of
the eddy currents. Certain earth formations, however, consist of
thin layers of electrically conductive materials interleaved with
thin layers of substantially non-conductive material. The response
of the typical electromagnetic induction resistivity well logging
instrument will be largely dependent on the conductivity of the
conductive layers when the layers are substantially parallel to the
flow path of the eddy currents. The substantially non-conductive
layers will contribute only a small amount to the overall response
of the instrument and therefore their presence will typically be
masked by the presence of the conductive layers. The non-conductive
layers, however, are the ones which are typically
hydrocarbon-bearing and are of the most interest to the instrument
user. Some earth formations which might be of commercial interest
therefore may be overlooked by interpreting a well log made using
the electromagnetic induction resistivity well logging instruments
known in the art.
[0009] U.S. Pat. No. 6,147,496 to Strack et al. teaches the use of
an induction logging tool in which at least one transmitter and at
least one receiver are oriented in orthogonal directions. By
operating the tool at two different frequencies, it is possible to
substantially reduce the effect of invasion and to determine the
orientation of the tool to the bedding planes.
[0010] U.S. Pat. No. 5,999,883 issued to Gupta et al, (the "Gupta
patent"), the contents of which are fully incorporated here by
reference, discloses a method for determination of the horizontal
and vertical conductivity of anisotropic earth formations.
Electromagnetic induction signals induced by induction transmitters
oriented along three mutually orthogonal axes are measured. One of
the mutually orthogonal axes is substantially parallel to a logging
instrument axis. The electromagnetic induction signals are measured
using first receivers each having a magnetic moment parallel to one
of the orthogonal axes and using second receivers each having a
magnetic moment perpendicular to a one of the orthogonal axes which
is also perpendicular to the instrument axis. A relative angle of
rotation of the perpendicular one of the orthogonal axes is
calculated from the receiver signals measured perpendicular to the
instrument axis. An intermediate measurement tensor is calculated
by rotating magnitudes of the receiver signals through a negative
of the angle of rotation. A relative angle of inclination of one of
the orthogonal axes which is parallel to the axis of the instrument
is calculated, from the rotated magnitudes, with respect to a
direction of the vertical conductivity. The rotated magnitudes are
rotated through a negative of the angle of inclination. Horizontal
conductivity is calculated from the magnitudes of the receiver
signals after the second step of rotation. An anisotropy parameter
is calculated from the receiver signal magnitudes after the second
step of rotation. Vertical conductivity is calculated from the
horizontal conductivity and the anisotropy parameter.
[0011] Co-pending U.S. patent application Ser. No. 09/676,097 by
Kriegshauser et al, the contents of which are fully incorporated
herein by reference, teaches a method for determining an applying
shoulder bed corrections to logging measurements made with a
transverse induction logging tool. Layer boundaries are determined
from the measurements. These are combined with horizontal and
vertical resistivities obtained by a whole space anisotropic
inversion to give a layered model. Preferably, a Lanczos iterative
procedure is used for the inversion. The shoulder bed correction
for each layer is derived based upon a difference between a 1 -D
synthetic response of the model and a whole space response of the
model at that layer. The shoulder bed correction is applied to the
data and the inversion procedure is repeated. This procedure is
repeated in an iterative manner until a difference between the
shoulder bed corrected measurements at the center of each of the
layers and a synthetic response to a whole space model at the
center of each of the layers is below a predetermined
threshold.
[0012] Kriegshauser teaches the use of a multicomponent induction
logging tool in which five components of the magnetic field as
shown in FIG. 1. This tool which is marketed under the name
3DEX.TM. by Baker Hughes Inc., measures three principal components
H.sub.xx, H.sub.yy, H.sub.zz and two cross-components H.sub.xy and
H.sub.xz. The measured data from 3DEX.TM. tool are unfocused and
thus inversion is necessary in interpreting the 3DEX.TM. data.
[0013] Only a few authors have attempted to invert borehole EM
logging data to resolve anisotropic formation. Gupta et al. and
Kriegshauser et al. have developed various inversion methods,
including whole space inversion, radial 1 D inversion, and vertical
1 D inversion, to resolve anisotropic formations. EM logging data,
however, are subject to borehole, shoulder, and invasion effects.
The drilling mud coupled with the high pressure during the drilling
process can create a flushed zone around the borehole that can be
as thick as several meters. The approximate correction for borehole
and shoulder effects using radial 1D and vertical 1D inversions
alternatively in Kriegshauser et al., works reasonably well in most
cases but these approximate corrections may not lead to
satisfactory solutions in complicated situations such as high
resistivity contrast and thin layers.
[0014] Additionally, different tools have different vertical
resolutions and they respond differently to the earth models.
Separate interpretations of multi-component induction tool and HDIL
data, therefore, can lead to different and sometimes inconsistent
results.
[0015] There is a need for a method of consistent interpretation of
HDIL and 3DEX data to obtain anisotropic resistivities of the
subsurface along with anisotropic resistivities of the invaded
zone. The present invention satisfies this need.
SUMMARY OF THE INVENTION
[0016] A multi-component electromagnetic logging tool is used for
obtaining multi-component measurements indicative of anisotropic
resistivities of the subsurface. An initial model of the
subsurface, including horizontal and vertical resistivities and
parameters of an invaded zone around a borehole, is defined. A
forward modeling program is used to obtain expected responses of
the multi-component tool based on the initial model. An iterative
procedure is used to update the model and minimize an objective
function related to the mismatch between the model output and the
multi-component measurements. The objective function also includes
a model objective function to stabilize the inversion process. The
initial model for the inversion may be obtained from other
measurements, such as an array logging tool.
[0017] In another embodiment of the invention, the data from the
multi-component logging tool are inverted jointly with data from an
array logging tool. This joint interpretation can give results
superior to those obtained from the inversion of the
multi-component data alone. The improvement is noticeable in
estimation of the invaded zone parameters.
BRIEF DESCRIPTION OF THE FIGURES
[0018] FIG. 1 shows an induction instrument disposed in a wellbore
penetrating earth formations.
[0019] FIG. 2 shows the arrangement of transmitter and receiver
coils in a preferred embodiment of the present invention marketed
under the name 3DEX TM
[0020] FIG. 3 shows examples of the response of some of the coils
of the instrument of FIG. 3 to an anisotropic earth.
[0021] FIG. 4 shows an example of the response of the 3DEX TM tool
to formation anisotropy.
[0022] FIG. 5 shows the results of using the method of the present
invention to invert data from the 3DEX TM tool.
[0023] FIG. 6 shows the results of using the method of the present
invention to jointly invert data from the 3DEX TM tool and an array
Induction tool.
DETAILED DESCRIPTION OF THE INVENTION
[0024] Referring now to FIG. 1, an electromagnetic induction well
logging instrument10 is shown disposed in a wellbore 2 drilled
through earth formations. The earth formations are shown generally
at 4. The instrument 10 can be lowered into and withdrawn from the
wellbore 2 by means of an armored electrical cable 6 or similar
conveyance known in the art. The instrument 10 can be assembled
from three subsections: an auxiliary electronics unit 14 disposed
at one end of the instrument 10; a coil mandrel unit 8 attached to
the auxiliary electronics unit 14; and a receiver/signal
processing/telemetry electronics unit 12 attached to the other end
of the coil mandrel unit 8, this unit 12 typically being attached
to the cable 6.
[0025] The coil mandrel unit 8 includes induction transmitter and
receiver coils, as will be further explained, for inducing
electromagnetic fields in the earth formations 4 and for receiving
voltage signals induced by eddy currents flowing in the earth
formations 4 as a result of the electromagnetic fields induced
therein.
[0026] The auxiliary electronics unit 14 can include a signal
generator and power amplifiers (not shown) to cause alternating
currents of selected frequencies to flow through transmitter coils
in the coil mandrel unit 8.
[0027] The receiver/signal processing/telemetry electronics unit 12
can include receiver circuits (not shown) for detecting voltages
induced in receiver coils in the coil mandrel unit 8, and circuits
for processing these received voltages (not shown) into signals
representative of the conductivities of various layers, shown as 4A
through 4F of the earth formations 4. As a matter of convenience
the receiver/signal processing/telemetry electronics unit 12 can
include signal telemetry to transmit the conductivity-related
signals to the earth's surface along the cable 6 for further
processing, or alternatively can store the conductivity related
signals in an appropriate recording device (not shown) for
processing after the instrument 10 is withdrawn from the wellbore
2. Turning now to FIG. 2, the configuration of transmitter and
receiver coils in a preferred embodiment of the 3DExplorer.TM.
induction logging instrument of Baker Hughes is disclosed. Three
orthogonal transmitters 101, 103 and 105 that are referred to as
the T.sub.x, T.sub.z, and T.sub.y transmitters are shown (the
z-axis is the longitudinal axis of the tool). Corresponding to the
transmitters 101, 103 and 105 are associated receivers 107, 109 and
111, referred to as the R.sub.x, R.sub.z, and R.sub.y receivers,
for measuring the corresponding magnetic fields H.sub.xx, H.sub.zz.
and H.sub.yy. In addition, the receivers 113 and 115 measure two
cross-components H.sub.xy, and H.sub.xz of the magnetic field
produced by the x-component transmitter.
[0028] FIG. 3 is a schematic illustration of the model used in the
present invention. The subsurface of the earth is characterized by
a plurality of layers 201a, 201b, . . . 201i. The layers have
thicknesses denoted by h.sub.l, h.sub.2, . . . h.sub.i. The
horizontal and vertical resistivities in the layers are denoted by
R.sub.hl, R.sub.h2, . . . R.sub.hi and R.sub.vl, R.sub.v2, . . .
R.sub.vi respectively. The borehole is indicated by 202 and
associated with each of the layers are invaded zones in the
vicinity of the borehole wherein borehole fluid has invaded the
formation and altered is properties so that the electrical
properties are not the same as in the uninvaded portion of the
formation. The invaded zones have lengths L.sub.x01, L.sub.x02, . .
. L.sub.x0i extending away from the borehole. The resistivities in
the invaded zones are altered to values R.sub.x01, R.sub.x02, . . .
R.sub.x0i . In the embodiment of the invention discussed here, the
invaded zones are assumed to be isotropic while an alternate
embodiment of the invention includes invaded zones that are
anisotropic, i.e., they have different horizontal and vertical
resistivities. The assumption of an isotropic invasion zone is
reasonable because in the case that the borehole fluid is
conductive and invades a laminated sand/shale layer, then the pore
fluid of the sand laminae is filled with conductive borehole mud
fluid. This results in the sand laminae becoming as conductive as
the shale laminae, thereby reducing anisotropy that would be
produced by an interbedded sequence of thin layers with contrasting
resistivity.
[0029] FIG. 4 shows the response of a model in which the layers are
anisotropic. The forward modeling we used in the inversion is the
one described in Tamarchenko and Tabarovsky. This forward modeling
algorithm takes advantages of the axially symmetric nature of 2D
borehole problem and adopts a fast hybrid numerical technique that
combines the integral equations and finite difference methods.
[0030] The curve 281 depicts the anisotropy ratio X (ratio of
vertical resistivity to horizontal resistivity) in the layers. The
actual resistivity values are not shown. The H.sub.zz in a vertical
borehole is insensitive to the vertical resistivity and is not
shown. The curves 251, 261 and 271 show the H.sub.xx response to
the model at frequencies of 21 kHz, 83 kHz and 222 kHz
respectively. Also shown by dashed lines are corresponding H.sub.xx
responses 253, 273 when there is no anisotropy in the layers. The
current flow produced by a horizontal transmitter cuts across
formation boundaries but also has regions where the current flow is
parallel to formation boundaries. As a result of this, the H.sub.xx
response depends on both the horizontal and vertical resistivities
and is also more susceptible to shoulder effects than the H.sub.zz
response. This, together with the effects of the invasion, results
in the situation shown in FIG. 3 where little effect of anisotropy
is seen on the H.sub.xx response. The most noticeable effect is at
a depth of approximately 250 ft. where there is a thick layer
(approximately 16 ft. in thickness) with a large anisotropy ratio
of 3: 1. This suggests that inversion of 3DEX.TM. data alone is not
going to give accurate results.
[0031] In induction logging, the sources are magnetic dipoles and
the secondary magnetic field is measured. In either case, the data,
D, can be expressed as a nonlinear function of the physical
parameter, m, via a nonlinear function, f
[0032] D=f(m) (1)
[0033] where m is a model vector that comprises the layer
thicknesses, the layer resistivities, and the length and
resistivity of the invaded zones.
[0034] The inversion goal is to find a model that reproduces the
data and exhibits the desired characteristics of the formation. The
data objective function, .o slashed..sub.d, is given by
.o slashed..sub.d=.parallel.W.sub.d(D.sup.obs-D).parallel..sup.2
(2)
[0035] where D.sup.obs and D are the observed and predicted data,
respectively, and W.sub.d is a weighting matrix for the data. If
the noise in the data components is Gaussian and independent, then
W.sub.d is a diagonal matrix whose elements are the reciprocal of
the standard deviation associated with each datum. In many
instances, the noise in the data will not be Gaussian independent;
in such a case, the covariance of the noise may be used to derive a
suitable weighting matrix W.sub.d.
[0036] As in all inversion problems, the success of the results
depends to some extent on the choice of the initial model. In a
preferred embodiment of the invention, the bed boundaries and
layers for initial model are derived using the short subarrays of
the HDIL tool and the 3DEX.TM. tool. The initial values for the
horizontal resistivities and the length and resistivities of the
invaded zones are obtained using the HDIL tool in a conventional
manner. The initial model is assumed to be isotropic, i.e., with
vertical resistivities equal to the horizontal resistivities.
[0037] The most challenging part of the 2-D inversion is the
appropriate handling of model parameters that have different
physical units and different effects on the data. In the present
invention, the choice of the model objective function is guided by
the desire to find a model that has minimum structure in the
vertical direction and at the same time is close to a reference
model. To accomplish this, model objective functions for
resistivity of the invaded zone and the length of invasion are
defined as 1 = w ln ( 0 ) 2 v + ( 1 - ) w ( ln - ln 0 ) z 2 v and (
3 ) l = l w l ln ( l l 0 ) 2 v + ( 1 - l ) w l ( ln l - ln l 0 ) z
2 v ( 4 )
[0038] where .rho..sub.0 and l.sub.0 are the reference models for
resistivity and invasion length. The integration is with respect to
vertical depth. In eqs. (3) and (4), .rho. includes the horizontal
and vertical resistivities as a function of depth as well as the
resistivity of the invaded zone.
[0039] The two parameters, .alpha..sub..rho.and .alpha..sub.l
control the relative importance of the smallest and flattest
components in the model objective functions. They are decided
automatically at each iteration by making the norms of eqs. (3) and
(4) the same. The use of In(.rho.) and In(l) ensures the
non-negative solution of the recovered models and allows the model
parameters for resistivity and invasion length to span the same
numerical range in the inversion. The two weighting functions,
w.sub..rho.and w.sub.l allow the user flexibility to incorporate a
priori information about model parameters into the inversion. The
discrete model objective functions, i.e., wherein the model
consists of discrete layers within which the resistivities and
invasion lengths are constant, can be written as 2 = ; W ln ( 0 )
r; 2 and ( 5 ) l = ; W l ( l l 0 ) r; 2 ( 6 )
[0040] where W.sub..rho.and W.sub.l are weighting matrices.
[0041] The model objective function for the inversion is then
defined as
.o slashed..sub.m=.eta..o slashed..sub..rho.+(1-.eta.).o
slashed..sub.l (7)
[0042] wherein the coefficient .eta.is given by 3 = 1 1 + s ( 8
)
[0043] where 0.ltoreq.s .ltoreq..infin. is the desired
magnification factor of the length in the inversion process
relative to the resistivity. Examination of eq. (8) shows that when
s is zero, the model objective function is the same as the length
resistivity objective function and when s is infinite, the model
objective function is the same as the resisitivity objective
function.
[0044] The global objective function for the inversion is a
combination of the data objective function from eq. (2) and the
model objective function from eq. (7). This may be written as
.o slashed.=.o slashed..sub.m+.beta..sup.-1(.o slashed..sub.d-.o
slashed..sup.tar) (9)
[0045] where .beta. is the Lagrangian multiplier and .o
slashed..sup.tar is the target misfit level (a noise factor that
specifies the acceptable misfit in the inversion process).
[0046] This nonlinear optimization problem may be solved using any
suitable method, such as Newton-Raphson or Marquardt-Levenberg. The
objective function at the n-th iteration is given by
.o
slashed.=.parallel.W.sub.m[.delta.m+m.sup.(n)-m.sub.0.parallel..sup.2+.-
beta..sup.-l{.parallel.W.sub.d{D.sup.obs-.function.[m.sup.(n)]+J.delta.m}.-
parallel..sup.2-.o slashed..sup.tar(n+l)} (10)
[0047] where m is the model vector comprising model parameters for
resistivity and invasion length (.rho. and l above) and
J=(J.sub..rho.,J.sub.l) are the sensitivities for resistivities and
invasion length respectively.
[0048] The global model weighting matrix is 4 W = [ W 0 0 1 - W l ]
( 11 )
[0049] In a preferred embodiment of the invention, the target
misfit level is reduced by a factor between 2 and 10 from one
iteration to the next.
[0050] The model is defined as a vector m m=[h.sub.l L.sub.x01
R.sub.x01 R.sub.hl R.sub.vl h.sub.2 . . . h.sub.n L.sub.x)n
R.sub.x0n R.sub.hn R.sub.vn].sup.T (12)
[0051] where .sup.T denotes transpose and where (h.sub.i,L.sub.x0i,
R.sub.x0i, R.sub.hi, R.sub.vi) denote the thickness, length of the
invaded zone, resistivity of the invaded zone, horizontal
resistivity and vertical resistivity respectively for the i-th
layer, there being a total of n layers in the model. In an
alternate embodiment of the invention, the thicknesses of the
layers are taken as fixed.
[0052] In one embodiment of the invention, the observations
comprise measurements made with the 3DEX.TM. logging tool
D.sup.obs=[H.sub.xxl H.sub.yyl H.sub.zzl H.sub.xyl H.sub.xzl . . .
H.sub.xxM H.sub.yyM H.sub.xzM H.sub.xyM H.sub.xzM].sup.T (13)
[0053] where the observations are made at a total of M depths. In
another embodiment of the invention, the observations used in the
iterative process also include the measurements made with the
HDIL.
[0054] From a practical standpoint, it may be desirable to perform
the iterations over a depth window centered over the observation
depth. This limitation becomes important only when constrained by
availability of memory and processing capability.
[0055] The iterative process requires the determination of a
Jacobian matrix J of partial derivatives relating elements of the
model vector to the elements of the observation vector 5 J = [ J 11
J 12 J 1 n J 21 J 22 J 2 n J M1 J M2 J Mn ] where ( 14 ) J ik = D i
obs m k ( 15 )
EXAMPLE
[0056] Turning now to FIG. 5, the results of using the method of
the present invention are shown. The shaded portion of left track
301 shows the model of the invasion zone that was used. The actual
resistivity of the invasion zone is shown by the solid line in the
track 311. The actual horizontal resistivity of the formation is
shown by the solid line in track 321 while the solid line in track
331 shows the actual vertical resistivity of the layers.
[0057] Synthetic data were generated by forward modeling using the
method given by Tamarchenko and Tabarovsky and 1 % random Gaussian
noise was added. Starting with an initial model in which the bed
boundary locations are known and fixed, and with an initial
isotropic model having a uniform resistivity of 1 .OMEGA.m for the
formation and for the invaded zone, the data were first inverted
using only the 3DEX.TM. data. The solid line in the track 301 is
the inverted length of the invasion zone, the dashed line in track
311 is the inverted resistivity of the invasion zone, the dashed
line in track 321 is the inverted horizontal resistivity while the
dashed line in track 331 is the inverted vertical resistivity.
[0058] FIG. 5 shows that the inverted values of the horizontal
resistivity R.sub.h agree quite well with the actual horizontal
resistivity model: there is little difference between the solid and
dashed lines in the track 321. The same is also true of the
inverted resistivity of the invasion zone in track 311. However,
the inverted vertical resistivity (dashed line in track 331) does
differ somewhat from the actual vertical resistivity. Similarly,
the invaded length of the invaded zone (solid line in track 301)
differs from the actual invaded zone (shaded portion in track 301).
Errors in the inverted length of the invasion zone are noticeable
at depths indicated by 345 while at depth indicated bn 341, a
fairly thick interval has some error in determination of vertical
resistivity. Nevertheless, using only the 3DEX.TM. measurements,
the inverted model does identify the zones with high anisotropy.
Identification of these anisotropic zones is of considerable value
in formation evaluation.
[0059] The process of inversion of the model output was then
repeated using all of the data, i.e., by including the 3DEX.TM.
data as well as the HDIL data. The latter data set would include
measurements using the HDIL data at seven different spacings and
eight different frequencies. The results of this inversion are
shown in FIG. 6. As in FIG. 5, the left track 401shows the invasion
length (shaded for true values, solid line for inverted values),
the track 411 shows the resistivity of the invaded zone (solid for
true values and dashed for inverted values), the track 421 shows
the horizontal resistivity (solid for true values and dashed for
inverted values) and the track 431 shows the vertical resistivity
(solid for true values and dashed for inverted values). FIG. 6
shows that the inverted values of invasion length and the vertical
resistivity are much closer to the true values than in FIG. 5 where
only the 3DEX.TM. data were used. In particular, the depths 345
shows considerable improvement in the inverted values of the
invasion zone length, and the depth range 345 shows a much improved
inversion of the vertical resistivity.
[0060] The method of the present invention has been illustrated
above using a simple model in which the layer boundaries are kept
fixed and the model includes invaded zones. The method has also
been tested to invert models in which the layer boundaries are also
allowed to be changed during the inversion process, and wherein
there are two different invaded zones at each depth. Such a
situation might occur if there is an invaded zone and a flushed
zone in the formation. Results have been comparably, showing that
the joint inversion of multicomponent (3DEX.TM.) and multiple array
induction tools (HDIL) gives results superior to those from
multicomponent tools alone.
[0061] The present invention has been discussed above with respect
to measurements made by a transverse induction logging tool
conveyed on a wireline. This is not intended to be a limitation and
the method is equally applicable to measurements made using a
comparable tool conveyed on a measurement-while-drilling (MWD)
assembly on a drillstring or on coiled tubing.
[0062] While the foregoing disclosure is directed to the preferred
embodiments of the invention, various modifications will be
apparent to those skilled in the art. It is intended that all
variations within the scope and spirit of the appended claims be
embraced by the foregoing disclosure.
* * * * *