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PETROPHYSICS. YOL44. NO, 3 (MAY·JUNE W 03); p, 196- 204, 9 FIGURES

Apparent Resistivity Correction for Tensor Induction Well Logging in a Deviated Well in an Anisotropic Medium

E rtan Pekscn I and Michael S. Zhdan ov'

ABSTRACT

In this paper we study the principles oftensor induction welI logging (TIWL) in a deviated we ll in an aniso tropic medium. We examine the theoretical fo rm ulae for the ten­sor appa rent conductivities of the transverse an isotropic medium, develo ped for an ideal tenso r induction instru ­ment with coinciding positions of three mutually orthogo­nal transmitters at one point and all three receivers at the other point in a borehole. We demonstrate that these for­mulae can be corr ected for prac tica l instru ment design ,

INTRODUCTION

Induct ion well logging in anisotropic forma tions has recently become an area of active research and industrial development. Kriegs hauser et al. (2000a, b) described the new induction inst rument com pris ing three mu tua lly orthogonal transmitter co ils and similar receiver coils. A number of papers discussing mul ti-component induction logging appeared at the 42nd meeting of the SPWLA in Houston, (Cheryauka and Zhdanov, 200 Ib ; Kriegshauser et al., 200 Ia; Marti n et al., 20 01; Zhang et al., 200 1; Zhdano v et al., 2001b) and at the 71st mee tings of SEG in San Anto­n io, in 2001 (Cheryauka and Zhdanov, 2001c; Yu et al., 2001; Kriegsauser et al., 20 0 Ib; Lu and Alumbaugh , 2001 ; Zhang, 2001; Zhang and Mezzatessa, 200 1).

The response of the tri-ax ial induction instrument is form ed by three components of the magnetic field due to each of three transmitters, thus form ing a tensor array of nine signals, which we ca lI an induction tensor. This tensor is determined by the conductivity tensor ofthe medium. The goal of interpretation is to recover the conductiv ity tensor from the observed data. The foundations of tensor induction well logging (TIWL) in an isotrop ic formations were out-

where, due to technical reasons , transmitter coils with dif­ferent orientations are spatially separated and rece iver coils are spatially separated as well. We introduce the cor­rected formulae for a practical TIWL instrume nt. The numerical study shows that, for different deviation ang les and for different anisotropy values, the corrected apparent conduc tivities are practi cally the same as the theoretical apparent parameters.

Manuscript received by the Editor March 28, 2002; revised manuscript received December 17, 2002. 'Universiry of Utah, Department of Geology and Geophysics, Salt Lake City,UT USA ©2003 Society of Professional Well Log Analysts. All tights reserved.

PETROP HYSICS196

lined in the paper by Zhdanov, Ken nedy, and Pekscn (2001a), where the authors derived the expressions for an apparent conductivity tenso r based on the low frequency asymptotic of the induction tensor. These formu lae were derived only for an ideal instrument that comprises three mutually orthogonal transmitter coils located at the same point and similar receiver coils located at another po int. In practice, however, it is technically difficult to place a1lthree transmitter coils at one poin t, or to place all three receiver coils at one point. That is why, in the practical instrument (for example, in the 3DEX instrument developed by Bakel' Atlas), the transmitter coi ls with different orientations are located at some distance from one to the other (Figure 1). A similar design is used for the receiver coils . In this situation the theoretic al formulae derived in Zhdanov et al., (2001a, b) for apparent horizonta l and vertical conductivities may produce an etTOl1COUS result.

In this work , we develop a method of apparent conductiv­ity correction to analyze practical induction instrument data . Throughout this paper, we call our hypo thetical instrument the "ideal instrumen t" and the typical design of the indus­trial instrument the "practical instrumen t." Using numeric al

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Apparent Resistivity Correction for Tensor Induction Wcll Logging in a Deviated Well in an AnisotropIc Medinm

simulation, we demonstrate that the correction method Hr .I'

introduced in this paper improves the accuracy of the Hf ]iI= ~ H{ (I)evaluation of the anisotropi c parameters of the medium by Hl' z; ,

H~ H~using the practical tri-axial induction data .

where superscripts refer to the transmitter components andCALCULATION OF THE APPARENT

subscripts refer to the receiver components.CONDUCTIVITIES IN TENSOR In the Cartesian sys tem of coord inates (x,y,z), with theINDUCTION WELL LOGGING

axis Z directed along the axis of symmetry of the trans­The ideal tensor induction well logging instrument versely isotropic (TI) conductive medium, the conductivity

detects three components of the magnetic field due to each tensor can be represented by of three transmitters for a total of nine signals, which form the magnetic induction tensor ail 0

oo],a= ~ o~[ a,.

where ail is the horizontal conductivity and a; is the vert ical conductivity. The mag netic field components are given in formula ( I) in a coordinate system defined by the horizont al J ' s" iFLIand vert ical principal axes of the conductivity tensor. In

\( i l practice, the orientation of the transmitter and receiver coils" will be arbitrary with respect to this coordinate system. In l~; . ll t l

• 1'1 order to use the representation of the field tcnsorH for an II Ii

1'1.'I~ , ,!,instrument having an arbitrary orientation with respect to ,1 :1'~

the principal axes of the conductivity tensor, it is necessary i; il!, to trans form the transmitter moment in the instrument frame~ . I" (denoted by (x', y ', Z')) into the medium coordinates (denoted ;;" ,1. by (x ,y,z )). This transformation can be made by application : ~ .'I I

I. \ " '.1, ~Tx of the rotati onal matrix R (Moran and Gianzero , 1979; '. 11

- - - - - - ~-- - - -" - - -- - - - - " - m 1.., -------------1 ~

\}

,.

,.

.,. ~

')

,.

,.

,. 'l'.

')

!· Zhdanov et al., 200 1a). The rotation matrixR cons ists of ., . .1(JfI -, Iitwo rotations >

-: '1 1L 2~ :; ;1 1.1R=R pR" , i:iii'L L_(1) _L <" IJ

where R describes the rotation around y' Ha

COS a 0 if'i-sin a]

'I: j

JI1:1

n, = 0 I o , sin a 0 cos a

[ "I , I

and Rfl describes the rotatio n around the z (= z') axis WIIi:l

through an angle [3 , and 'I.'.. f~

COS f3 sin f3 iJJ 1 A ... Rx R,B = -s~nf3 cos f3 . ·11'1j

[ , Ino L: f ·1

FIG. 1 A simplified practical instrument has three mutually where a represents the relative deviat ion angle between the ' ...I 'j'> ortogonal transmitters and three receivers as an ideal instru­ . ', r ~ i

ment does, but they are separated from each other. ins trument axis and the "verti cal" principal axis of the con- ~

e­

197May-June 2003 PETROPHYSICS ~

"l'"

~ f : IU et

t, :Pcksen and Zhduuov

ductiv ity tensor, and f3 represents the rotation of the instru­

ment on it ow n axis.,I The product ofthese two rotation matrices is given by

COS a cos f3 sin (3 - sin a cos 13. R==RpR"= -co~ a sin (3 cos(3 sin a sin (3 . (2)

[ SIn a 0 cos a

The matrix of the magnetic induction tensor in the instru ­ment frame is related to the mat rix of the magn etic induc­tion tensor in the medium coordinate fra me by the foll owing form ula:

fr :== RirR - == RtIRT . (3)I

Ii

,~ .- !' The components of H'are used in the estim at ion ofreceiver

volt ages. App endix der ives a low- frequency approximation for the.I tensor induction magnetic field components and giv es for­

mulas for th e apparent horiz ontal and vert ical conductivi ­ties, aT,., and or", the app arent anisotropy coefficient, ),1" and the apparent relative deviation angle, a~; , for the ide al instrument.

\ "

APPARENT CONDUCTIVITIES FOR THE PRACTICAL INSTRUM ENT

A simplified des ign of a typical practical instrument is shown in F igu re 1. For this instrument, all three tran smi tter coils and all three receiver coils are placed at different po ints along the sond e ax is . However, the dist an ces bet ween the transmitter and receiver co ils with the same ori­entations are the same , L 1 , while the distances between cross oriented transmitters and receivers arc different (see Figure 1). For exam ple, the distance between the x-oriented trans­mitter dipole and the z-oriented receiver dipole is L2 ;c L I •

The analys is of formulae (A .6), (A .7), (A .8), and (A. 9) shows that th e expressions for apparent parameters (hori­zontal and vertical conductivities, anisotropy coeffici ent, and relative deviation angle) contain only tour components of the magnetic indu ction tensor: H~:, H::::, H~:, and H;:. We should recall now that these formulae we re derived in a homogeneous TI med ium . It is clear that in a homogeneous TI me dium the components dep end only on the distance between the tran smitt er and receiver, and they do no t depend on the relative loc ations ofthe transmitters along the ., sonde axis. Therefore, the components, H:~' , Hr ,and H;:areI the same fo r the ideal and practic al instruments,

_i ~ . At the sam e time, the H;:component is different for the ~ ". ideal and p ractical instruments, because the distance L 1'r. .~ , betw een the x-oriented transmitte r dipole and the z-oriented .~ " receiver dip ole for the ideal instrument is different than the

'l1

distanc e L2 for the practical instrument. However, from for­mula (A .3) we see that the H;': comp onent is inversely pro­ <:>

po rtional to the transmitter-receiver separation. As a result, <.l

we have

(<;SH :;:) = ( ~H :;:) L2, (4) \ :

• L, - L, LI

~. ,

where ( ~H~' :)LI L, is the imag inary magnetic fi eld compo­nen t w ith the transm itter-receiver sepa ration equal to L J or on

L2, respec tively. ,­Thus, we can use theoreti cal formu lae (A.6 ), (A ,7), ..

(A.8), and (A.9)for compu ting the apparen t parameters for

the practical instrument dat a, if we replace (C:Sf n :)LI with ~

the expression (~H;:)dL2/L I ) acc ording to equation (2), l, The new corrected formulae for the apparent parameters computed for the practical instrument are

ah" = , 1 ,

(CSH\~)L + - (CSH;' )L + . , 2 ­ 11

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.. ~

0 0 "2go , :.' 1 .' - 0' L2 ­( ~W, ) - - (~fJ::. ) ) + 2(~'- H:,) (-), I( x L, 2 - L, • . L, L

1

(5) ·t

..' L' _ 1 . 2 (CSH ~:)L , (L2 ! LI) ]-l[

a a --sm C'x x' =' c ' (6) 4 J2 (osH ,,' )L, + (CSH / )L, -3g0a ha

4.~where ahais the corrected horizontal apparent conductivity, , and a~; is the corrected apparent relative deviation angle. The

<!,subscr ipts L1 and L2 corresp ond to transmitter-receiver dis ­tances and indicate that the magne tic field is measured or

~ calculated at these distances. Equat ion (A.7) does not need to be modified since this formul a does not contain the H;:

~:component. Once the horizontal apparent conductiv ity is calculated, the ver tical conductivi ty may be obtained from equation (A.9): <':

c c . "I)a ha

(7)Ova - ().~, )2 ~\

NUMERICAL EXAMPLES <,

In this section we examin e the accu racy of the corr ected I

formulae for the apparent parameters, using numerical -€'

!

simulation . Fo r simplicity, we consider the tra nsverse ani ­sotropic model of the rock formation without borehole and *' inva ded zones. We do not include the borehole effect in the ,jJ

modeling study since the borehole correction is usually per­

4

PETROPHYSICS May-Jun e 2003 198 11

fijI>

Apparent Resistivity Correction for Tensor Induction Well Logging in a Deviated Well in a n Anisotropic Medium '",!J.I! ." ,r.( l ~ !I

'j ~ " (. tformed for induction tool data before any other interpreta­ resis tivity curves for the ins truments with the relative devia­-~~ G 1"

The distances betwe en the transmitter posi tions and cal apparent res istivity curve computed for the practical tion. tion angle a = 10° are g iven in Figure 3. Note that the verti­

~ f: " l ~ the rece ive r pos it ion s in th e practical ins trument are LI == instrument data by the th eoretical formula derived for the

1.0 m and L2 = 0.40 m. Th e operating frequency range is up ideal instru ment becomes negative (dashed line in Figure 3, mH to 30 kHz. The model parame ters are the horizontal resistiv­ bottom panel), while after the correction we obtain reason­" ,r;dIr" ity 10 ohm-m, the verti ca l resis tivity 160 ohm -m, and the able values for the vert ica l apparent conduc tivity (dashed ;PI

~ anisotropy coefficient is 4. The relative deviation ang les are line in Figu re 3, top pan el), coi nci ding with the curve for the selected equa l to 10° and 30°. ideal instrument (solid line) . Figures 4 and 5 disp lay similar ,t i

' ,:, 1t·' l·' ·f..We computed the synthetic data for ideal and pract ical resu lts for the ide al and pra ct ica l instruments with the rela­induction instruments, and applied the corrected formu lae tive deviat ion angle a = 30 °. In this case, we correct our ~. ~

~,

(5) and (7) to the synthetic data obtained for the practical prac tical instrument dat a very well for the horizontal appar­instrument wi th the re lative devia tion angle a =10°. Figure cnt resistivity and re latively we ll for the vert ical resistivity.

• 2 presents the results of computing the hori zon tal apparent The co rrec ted apparent ver tica l resistivity for the pract ical resistivity curve s for the idea l instrument (sol id lines) and and ideal instru ments di verge sli ghtly with the increa se of

;. for the pract ical instrumen t (dashed lines ). The bottom part the fre quency (Figure 5 , top pa nel). of Figure 2 shows the compari son of the apparent resistiv i­ In the next set of numer ical experiments we invest igate ll;> ties calculate d for the idea l and practica l instruments wi th­ the behavior ofthe apparent res istiv ity curves in the layered fi l out a correction fact or (by using formula (A.6), while the anisotropic formations . The dis tance between the transmit­

~ top part presents the same results obtained for the practi cal ter 's position and the receiver 's position is 1.0 m for the IE

:i"i" instrument wi th the corrected form ula (5). One can see that, ideal ins trument, and th e transmit ter-receiver separations

:-after correction, two apparent resistivity curves generated for the practical instrument, shown in Figure 1, are as fol­ It ~ 'fj.

I t1 ;H ;for the ideal (solid line) and practical (dashed line) instru­ lows : L I = 1.0 m and L2 = 0040 111 , The calculations are per­ I II ·h·;

~ t\ :: tment responses are almost identical. The vertical apparent for med for an instrument m oving along the boreho le I, ,1 ;'1

'>

). : 4 u: 10 (With corre ctio n) :> 20,,-- ----- - ,.-- - - - - ---r----------,

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f -=­ IdealI - - - Practical

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':> 15f­ ;H ~

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"l, i!~ '..! !~l00:t - - --J- t I ~ .."----- ------ ------ ----- -----II -- Ideal

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o 1 2 3 ::> x 10' Frequency (HZ) ,Frequency (Hz)

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::> FIG. 2 An anisotropic model with the horizontal resist ivity 10 FIG. 3 An an isotropic model with the horizontal resistivity 10 l ~ lohrn-rn and the ve rtical resistivity 160 ohm-m. The relative ohm-m and the vertical resistivity 160 ohm-m. The relative deviationangle is 10°. Panels show horizontal apparent resistiv­ deviation a ngie is 10°. Pa nels show verticalapparent resistivity 11':'1> itycurves for the ideal instrument (solid li nes) and for the practi­ curves for the idea l instrume nt (solid lines) and forthe practical cal instrument (dashed lines).The bottompanel shows the com­ instrument (dash ed lines) , The bottom panel s hows the com­ Ii.:: ··,..·1. '> parison of the apparent res istivities calculated for the ideal and parison of the apparent res istivities calculated for the ideal and ~ /

practical instruments without a correction, while the top pane l practical instruments without a correction, while the top panel t!

1\·'.···

( ' :> presents the same results obtained for the practical Instrument presents the same resu lts obtained for the practical inst rument );:! with the correction. with the correction,

r> 'lt,i.',!May-J une 2003 PETROPHYSICS 199.. Hi r .l ~ I' ! :til

3

x 1 0~ H ~ l (Without cor rection)

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i t:1Peksen a nd Zhda uOY

upward with measurements every 0.25 m. The operating frequency is 20 kHz.

The synthetic data were computed using the OT3D code developed by the Consortium for Electromagnetic Model­ing and Inversion (CEMI) at the University of Utah (Cher­yauka and Zhdanov, 200 la) . The response of the layered anisotropic model was calculated in the model coordinate system. The result then was transformed from the model frame to the instrument frame, using equation (3).

Weassume that tensor induction logging is conducted by an instrument oriented along the borehole so that the "verti­cal" transmitter and receiver are oriented along the bore­hole. We calculate the model responses (induction tensor components for the ideal and practical instruments) for the different parameters of the layered anisotropic models. Using these responses as the synthetic observed data, we compute the low frequency (ara and a :'a) apparent conduc­tivities according to the theoretical formulae (A.6) and il (A.9) for the ideal instrument, The apparent conductivities, ar" and a~" for the practical instrument are computed using the corrected formulae (5) and (7) as they are described in I

j

f the previous section. The results are plott ed as apparent

I resistivity curves, Pre.. p;,;" p,;", and P ~lJ ' versus depth for a different relative deviation angle a , equal to 300 and 600

,

respectively (Figures 6 through 9).

~: 4 C1. : 30 (WIth correction) 20

1

15 - ,

~J :~ 6f

., oLI -:- _ ] -, J o 3

4

!X 10

(With ou t cor rectio n)

20 [ -- Ideal Practical . j - -­

I 15

;l- -- - -- -u -:- - - - - - - - - --~ - - - --- - - --- l!! °0 3

L Frequency (Hz) X 10· r FIG. 4 An anisotropic model with the horizontal resistivity 10 ohm-m and the vertical resistivity 160 ohm-rn, The relative deviationangleis 30°. Panels show horizontal apparent resistiv­itycurves for the ideal instrument (solid lines) and for the practi­calinstrument(dashed lines).The bottom panel shows thecom­parison of the apparent resistivities calculatedfor the idealand practical instruments without a correction, while the top panel presents the same results obtained for the practical instrument with the correction.

i;'

Each of the Figures 6 and 7 has three panels. The first panel on the left shows the parameters of the layered model.

r ., <:"

The second two panels present the apparent resistivity curves versus depth. The bold solid lines show the true ~)

parameters of the model. The apparent resistivities P;'n and p;,:,obtained by the theoretical formulae for the ideal instru­ 1,'j

ment are shown by the bold solid-dotted lines. The black solid-dotted lines represent the apparent resistivities com­ 4:, ~

puted for the practical instrument by the theoretical formu­lae (A.6) and (A.9) without correction. The gray solid­ ~"' dotted lines show the same parameters computed for the

.Ii "!practical instrument data after corrections. Solid lines describe the positive values of the apparent resistivities, i ....while the dotted lines correspond to the negative values.

The data were contaminated by 3% random Gaussian .t.;;.~noise at each transmitter-receiver position. Note that, for a

vertically oriented instrument, there is no need for correc ­"'~tion because the induction tensor component J-r;: is equal to

zero. In a deviated borehole we observe significant differ­ences between the bold solid-dotted and gray solid dotted ""r lines, which correspond to the apparent resistivities com­

"':puted without and with the correction, As a rule, after the correction the apparent resistivities computed for the practi­

tl,cal instrument coincide with the theoretical apparent resis­tivities for the ideal instrument. Thus, we can conclude that

600

400

E 200

9­0~

·200

-400 0

600

1400

l =L ~ J 0

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-400 ' o

i)!

!:: A : 4 c : 30 (With correction)

--­ ----­ --­ - --- ­ "'1

-------­ i, "";

1:-::­ Ideal l Practical t

i .;. . 1

2 3 ~X 10" ,

(Without correction) ,

Ideal l 41r:-::- Practical

1 ! ~

(l

I 3 4; Frequency (Hz) X10·

FIG. 5 An anisotropic model with the horizontal resistivity 10 ~ ohm-m and the vertical resistivity 160 ohm-m. The relative deviation angle is 30°. Panels show vertical apparent resistivity ,f/

I

curvesforthe ideal instrument(solid lines) and forthe practical instrument (dashed lines). The bottom panel shows the com­ 1

~ parison ofthe apparent resistivities calculated for the ideal and practical instruments without a correction , while the top panel I

prese nts the sa me results obtained for the practical instrument iJ. with the correction.

~l

PETR OPHYSICS 200 May-June 2003 £~

(!I!ldr'!'ll f 'iiil

hf:

j P,Apparent Resistivity Correction 1'01' Tensor Induction Well Logging in a Deviated Well in nn Anis otropic Medium

rlil l l~- 1

the corrected formulae work extremely well for all these tion instrument with co inciding positions of all three trans­ ~i ~ models. mitters and coinciding positions of all three receivers, can

We also considered a multi-layered model of anisotropic be corre cted for a practical instrument design. We have m formations , based on the known benchmark "Chirp" model introduced the corrected formulae for a practical TrWL

fh

f, . l J~ l~(Fang and Wang, 2000). This model is represented by the instrument. The numerical study shows that, for different ;~ r

black solid line in the left panel of Figure 8 (the horizontal relative deviation angles and for different anisotropy val­ lfil resistivity profile). We modified this model, adding a pro­ ues, the corrected apparent conductivities are practically the !: ; 1"I.....i·I·'·....."Ifile of vertical resistivity (right panel in Figure 8), The same as the theoretical apparent parameters. This result ;: ;1results of the synthetic TIWL data interpretation for differ­ indicates that the basic principles of the tensor induction ent relative deviation angles 30° and 60° are shown in Fig­ logging developed by Zhdanov et al. (2001a, b) for the ideal .~.I ' ures 8 and 9. Once again the apparent resistivities describe tensor instrument under low frequency condition can be I:' '

~;, ;;well thehorizontal resistivity and recover relatively well the easily modified for use with pract ical tensor instrument L :; vertical resistivity. These results show that, in the case of data. \ ~~ 'I'::t ,. i~

h,.complicated geoelectrical models, the simple interpretation tool based on apparent resistivity can be used. ~r'

ACKNOWLEDGMENTS L 1 ~

The authors acknowledge the support of the University ~,

CONCLUSION I-of Utah Consortium for Electromagnetic Modeling and i

In this paper we have demonstrated that the theoretical Inversion (CEMI), which includes Baker Atlas Logging > 11f01111Ulae for tensor apparent conductivity of a transversely Services, BHP Billiton World Exploration Inc., Electro­ '':r

anisotropic medium, developed for an ideal tensor indue- magnetic Instrumen ts , Inc., Exxo nMobil Upstream

Model 0 , i

> \ - mode l

- P:. ..... P

Ctli>

5 Sf Ii - P::' - po3 i\1Y1I

a

5

noglll iva .. 10f~ 10 10 r~ i>

=20 kHt T-A=1 m h ze 0.25 m

1(J.:30degrce s

1Ph1 =1 n-rn PV1=4 n-m

Ph2· 5 n - m Pya= 20 n - m h2:>5 m

PIt:l= 1n-m ,por,I = 4 n - m

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Research Company , INCO Explorat ion, International II -I 1 t. ' ; , ~ ' I...'. ;1. ;Energy Se rvic es, Japan Nationa l Oil Corporation, I I "jl'

MIND ECO, Na val Research Laboratory, Rio Tinto­ I I (i Kennecott, Shell Internat ional Exploration and Production ij,'j' ~

' , ' .Inc., and Sumitomo Metal Mining Co. {;':

,~ - P~II 1;­t"O,.,' C

'11The authors are also thankful to Dr. Leo Tabarovsky of

J _ . PII"

,; - f1 ~. Baker Atlas Logging Services and David Kennedy of Exx­ i{- p05lllv. ' ~ · .'. ",...." ,

," ~-.' Model

0 f:;:20 kHz

T- R=1 m t\ z= 0.25 m (.t = 60 dagreBs

r 5 f

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-=Q. Cll

PM = 1U- m Pvt = 4 0 -m

P~2 :O;: 5 !'l:-m PV2 :::200-m h2=S m

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()V3 =4 Q-m

25L-U 'C 15

0

5

10

15

FIG. 6 The first panel on the left shows a three-layer model. 20 1 1 2Df > The secondand the third panelsdisplay the horizontaland verti­

cal apparent resistivity curvesversus depth, respectively. Solid > lines describe the positive values of the apparent resistivities,

a - modol

- [J~II ."... P;tl

5- fl ~. - pofJIlv.

tlGQa ltvl

10

15

I I 1 20f II l H;'.:.H' '. il

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3- model : '...!.~.l- P: '~ H

i: P~ '; p '11 .-;- PYa I~ :- IlC~~ ~tI

;1t { , . tl8(l": lvD 'I" ./ lIt,

:,'1;,. ~ t ~

I!':, 'i!i 'l~ ;::\

25 L--.a! 25L.---JW :;. 1while the dotted lines correspond to the negative values. The

25 ' 10-1 10' 10' 1Q-1 10' 10'

.

t', i

Ph. (n - m) p,,(H- m) > relative deviation angle is 30°. The bold lines represent the

apparent resistivitiesp h8andp ra obtainedbythe theoreticalfor­ 1';1 mulae for the ideal instrument. The black solid-dotted lines rep­ FIG. 7 The fi rst panel on the left shows a three-layer model.

> '.'resent the apparent resistivities computed for the practical Thesecond and the third panels displaythe horizontaland verti­instrument by the theoretical formulae without correction. The cal apparent resistivity curves versus depth, respectively. The ';:ii i"1

d '1 '1> grary solid-dotted lines show the same parameters computed relative deviation angle is 60°. The curve's code isthe same as ;'..'1

fo rthe practical instrument data aftercorrections. in Figure6. > in!

~ i f i !~ <:i

May-June 2003 PETROPHYSICS 201 l>

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onMobil Upstream Research Co . for the stimulating discus­sions which helped to initiate this project.

REFERENCES

Cheryauka, B. A. and Zhdanov, M. S., 200 la , Electromagnetic tensor Green's functions and their integrals in transverse iso­tropic layered media: Proceedings of the CEMI 2001 Annual Meeting, p. 39-82 .

Cheryauka, B. A. and Zhdanov, M. S., 200 Jb, Fast modeling of a tensor induction tool response in a horizontal well in inhomo­geneous anisotropic formations, paper P, in 42nd Annual Log­ging Symposium Transactions: Society of Professional Well Log Analysts.

Cheryauka, B. A. and Zhdanov, M. S.. 200 Ic, Focusing inversion oftenSOl' induction logging data in an anisotropic formation and a deviated well, in 71st SEG Annual International Meeting Expanded Abstracts: Society of Exploration Geophysicists, p. 357-360.

Fang, S., and Wang, T., 2000, Accurate Bom simulation of indue­lion response using an optimal background. in 70th SEG Annual Intemational Meeting Expanded Abstracts: Society of Exploration Geophysicists, p. 1806-1 809.

Kriegshauser, B., Fanini, 0., Gupta., P., and Yu, L.. 2000a, Advanced inversion techniques for multicomponent induction log data, in 70th SEG Annual International Meeting Expanded Abstracts: Soci ety of Exp loratio n Geophys ic is ts, p. 1810-1 813.

Kriegshauser, B., Fanini, 0 ., Forgang, S., Itskovich, G. , Rabino­vich, M., Tabarovsky, L., Yu, L, and Epov, M.• 2000b. A new

:p!."" ! "::f.d se} \4"----",.& ... . ... i 50 •&,"" r~ P;a

rt . fI? : :o.! : z:

1001- II nU9 100 neg

150 ~ 150

§:€~

..::> I ~~

~ ~ .., ..:::::» .., 250 250

~OO I ~ 1 ~oo l . I

::~ ~5;i .aso

o , 2 e 4 400 - ­10 10 10 10 10 10' 10' 10' 10' 10'

Ph.{(l - m) p,, ((l - m)

FIG.B A generalized chirp type model of anisotropic formation (Fang and Wa ng, 2000) . The left and the right panels display the horizontal and vertical apparent resist ivity curves versus depth, respectively. The relative dev iation angle is 30°. The curve 's code is the same as in Figure 6.

'::"

multicomponent induction logging tool to resolve anisotropic formation, paper D, in 41st AnnualLogging Symposium Trans­ ; actions: Society of Professional WellLog Analysts.

Kriegshauser, B., Fanini, 0. , Yu, L., Horst, M. and Popta, J., 1:';

2001a, Improved shale sand interpretation in highly deviated and horizontal wells using multicomponent induction log data, v paper S, in 42nd Annual Logging Symposium Transactions: Society of Professional Well Log Analysts.

~~: ~

Kriegshauser, B., McWillams, S., Fanini, 0 ., and Yu. L., 2001b, An efficient and accurate pseudo 2-D inversion scheme for multicomponenet induction log data. in 71st SEG Annual Inter­ ~.;:

national Meeting Expanded Abstracts: Society of Exploration Geophysicists. p. 369- 372. (~

LtI, X.and Alumbaugh, D. L., 2001,One-dimensional inversion of three-component induction logging in anisotropic media, ill ....

r71st SEG Annual International Meeting Expanded Abstracts: Society of Exploration Geophysicists, p, 376-3 80. ~;

Martin, L., Chen, D., Hagiwara, T., Strickland, R., Gianzero, S., and Hagan, M., 2001, Neutral network inversion of array .t~-;

induction logging data for dipping beds, paper U, in 42nd Annual Logging Symposium Transactions: Society of Profes­ I sional Well Log Analysts. l'

Moran, 1. H., and Gianzero, S. C., 1979, Effects of formation ani­sotropy on resistivity logging measurements: Geophysics , vol. r-~ 44,p. 1266- 1286.

Yu, L., Kriegshauser, 8. , Fanini, O. and Xiao, 1., 200I, A fast "= I

inversion method for multicomponent induction log data, in 71st SEG Annual Intemational Meeting Expanded Abstracts: ~j Society of Exploration Geophysicists, p. 361-64 .

iZhang, Z., Yu, L., Krieghauser, B. and Chunduru , R., 2001, Simul- 'i:

"'l

Ia::: 60 degrees ; .('4

: {.Od I ' ~. ~d 1 ~50 ... Ph~ ] 50 p~.

Pho -- pes :

P t: 1

100 . . . n&y 100 ' " Ileil l 4 !

r<

, 150 150 4.'

:[ §:£& 5 ~ ~ ~ ~ ..,

250 250 I, "'rl

300 ~ 300 I ~ ,:::1 i . . sse Ij

<I!: e 1 2 400

10 10 10 10' 10' 10' 10' 10' 10' 10' Ph.((l- m) pv, (U- m) i

.(,,

(FIG.9 A generalized chirp type model of anisotrop ic formation I

(Fang and Wang, 2000).The left and the right panels display the ~

horizontal and vertical appa rent resistivity curves versus depth. ! respectively. The relative deviation angle is 60°. The curve's ~

code is the same as in Figure 6.

~

PETROPHYSICS May-June 2003202 4J

...:..

Apparent Resistivity Correction for Tensor Ind uction Well Loggi ng ill n Deviated We ll in an Anisotropic Medi um

taneousdetermination of relative angles and anisotropic resis­tivity using multicomponent induction logging, paper Q, in 42nd Annual Logging Symposium Transactions: Society of Professional Well Log Analysts.

Zhang, Z. and Mezzatesta, A., 200I, 2D anisotropic inversion of multicomponent induction logging data, in 71st SEG Annual Intemational Meeting ExpandedAbstracts: Societyof Explora­tion Geophysicists, p. 365-368 .

Zhang, Z., 200I, Recovering absolute formation dip and ID ani­sotropic resistivity structure from inversion of multicomponcnt induction loggingdata, in7 1st SEG Annual International Meet­ingExpandedAbstracts: Society of ExplorationGeophysicists, p.373- 375.

Zhdanov, M. S., Kennedy, D., and Pcksen, E., 200Ia, Foundations ~ of tensor inductionwell-logging: Petrophysics , vol.42, no. 6, p.

588-610 . Zhdanov, M. S., Kennedy, W. D., Cheryauka, B. A, and Peksen, E.,

2001b, Principles oftensor induction well logging in a deviated well in an anisotropic medium, paper R, in 42nd Annual Log­ging Symposium Transactions: Society of Professional Well Log Analysts.

AP PEN DIX

LOW FRE QUENCY APPROXIMATION FOR THE TENSOR INDUCTION

MAGNETIC FIELD COMPON ENTS

It was demonstrated by Zhdanov et a!. , (200 1a) that the effect of sonde roll can be evaluated from the observations

> of data in unbounded, homogeneous TI anisotropic medium, so that the rotational angle f3 can be taken equa l to zero. Under this assumption the express ions for low fre ­quency asymptotics ofthe induction tensor quadrature com­ponents in the instrument coordinate fram e are

r:sH~~' = go a,, [ 1 + 2q cos 2 a J, (A.I )

100 8iHj~ :' =go a"[ .J 2 - 2q - I] , (A.2) A. sin 2 a + ,1,2 cos 2 a

8iH~; = go a" [2q cos a sin a] = 8iHi ' , (A.3)

8iH:: = goa,,[2+ 2q sin 2 a], (AA)

>

>

100

>

100

>

A. Jsin 2 a + ,F cos 2 a - A. (A.5) q = Asin 2 a

8i denotes an imaginary part of the magnetic field compo­nent; H:~:, Hr , Ht,and H~: are the magne tic field compo­nents in the instru ment coordinate system; go is a parameter,

illjJ. O

g o = 81CL '

ill is the angular frequency.zz., is the free-space magnetic per­

meab iliry. Z =.J a" / a" is an anisotropy coefficient, and Lis

the transm itter-receiver separation. The analysis o f the low frequency as ympto tic s

(A. I )-(A.4) leads to the formulae for the low frequency "horizontal" apparent conductivity, aT"" the apparent ani­sotropy coefficient, A~, and the apparent re lat ive dev iation angle, a ~, given by (Z hdanov et al . 2001a, b),

T _ a ha ­

~( ii,I ..' I -v, . ' .' I , ., , .'

- rsH~, +- SsH::, + \SH'. - - ~a-J;, 2 +2 \) -H~ , ~ , ~ I2g() [ .\ 2 - ,\ 2 - ) ­ I.

t

(A.6) Ii, 11:

;; , A. ~ = , .. 4g a(~ J,:,) 2 , ,(A.7) C''sl-{ =, ( ("orH'\' + 8,H'~ + 0,' l-J=, - 2"0a T ).... = .... y .v ...... . z 0 1 lUI It

« .

T 1 . I 2 8iH_ ~: ]- - 1- I , , (A.8)

all - 2 Sl1 [ '2sH:- + \sH:, - 3g aL o

where superscript T indicates that these are the theoretical parameters computed for the "ideal instrument."

The expression fo r the "vertical" apparent conductivity, T • am IS

T T "aim

(A.9)a i; - (A.~) -

Thus, the TIWL method consists in measur ing the com­ponen ts of the magnetic induction tenso r us ing a tri-axial induction instrument and computing the apparent conduc­tivit ies using formulae (A.6) and (A.9).

May-June 2003 PETROPHYS ICS 203 >

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Peksen and Zhdanov ~~) ..

ABOUT THE AUTHORS 'f>

i

Michael S. Zhdanov is Professor of Geophysics , Director of

L the Consortium for Electromagnetic Modeling and Inversion (CEMI) at the University of Utah since 1993 , He received his PhD in physics and mathematics in 1970 from Moscow State Unive r­sity, He was previously the Director of the Geoelect rornagnetic Research Institute ofthe Russian Academy ofSciences, Moscow.

Dr. Zhdanov is Honorary Gauss Professor, Gettingen Academy of Sciences, Germ any, Full Member ofRussian Academy of Natu­ral Sciences, Hono rary Professor ofChina National Ce nter of'Geo­logical Exploration Technology, and Member of the Electromag­netics Academy, USA.

His main research inter est is developing methods for the solu­l l :ii! tion of forward and inverse problems in geophysics, new tech­! ljo ' niques for electromagnetic ima ging ofthe earth 's interior. He is thef. t :i author ofseveral books on geophysical elec tromagnetic theory and

inverse prob lem solut ions .

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i..Ertan Peksen is a PhD student at the Consortium for Electro­ r

magnetic Modeling and Invers ion (CEMI), the University ofUtah. He graduated in 1993 (BS) from the University of Ankara, in r

geophysics. He received his Master of Science degree in geophys­ I I

ics from the Department of Ge ology and Geophysics, University lil of Utah, in 1999. !

His research interest is modeli ng and inversion of tenso r induc­ i:l tion well logging data in anisotropic media.

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