Effect s of Geo met r i cal, Envi ron m en tal , and ... · August 2007 PETROPHYSICS 277 PETROPHYSICS, VOL. 48, NO. 4 (AUGUST 2007); P. 277–294; ... exam ine the sen si t iv ity

Effects of Geo met ri cal, Envi ron men tal, and Petrophysical Param e ters onMulti-Com po nent Induc tion Mea sure ments Acquired in High-Angle Wells1

Rob ert K. Mallan2 and Carlos Torres-Verdín2

INTRODUCTION

Pres ence of mac ro scopic elec tri cal ani so tropy, such asthat result ing from thinly lam i nated sand-shale andsand-sand sequences, can sub stan tially affect con ven tionalbore hole induc tion mea sure ments and cause an erro ne ouseval u a tion of hydro car bon sat u ra tion. The addi tion of

non-sym met ric and non-uni form mud-fil trate inva sion

result ing from high-angle and hor i zon tal wells pres ents an

addi tional degree of com plex ity that is hard to dif fer en ti ate

from other envi ron men tal and petrophysical effects.

Multi-com po nent induc tion tools are now avail able that

August 2007 PETROPHYSICS 277

PETROPHYSICS, VOL. 48, NO. 4 (AUGUST 2007); P. 277–294; 17 FIGURES

ABSTRACT

This paper describes a numer i cal study exam in ing theeffects of geo met ri cal, envi ron men tal, and petrophysicalparam e ters on multi-com po nent elec tro mag netic (EM)induc tion log ging mea sure ments. Direct and cross-cou -pled induc tion mea sure ments pro vide direc tional infor -ma tion on dip ping and non-uni form struc ture, and enablethe esti ma tion of resistivities par al lel and per pen dic u lar to res er voir lay ers. How ever, geo met ri cal, envi ron men tal,and petrophysical effects can sig nif i cantly affect thesemea sure ments. Under stand ing the cor re spond ing bias ingeffects aids in the inter pre ta tion of induc tion mea sure -ments and sub se quently leads to more accu rate and reli -able for ma tion eval u a tion.

We per form numer i cal sim u la tions of multi-com po -nent induc tion log ging mea sure ments with a 3Dfinite-dif fer ence mod el ing code. A suite of mod els is con -sid ered, includ ing a lay ered res er voir with vari able con di -tions such as bore hole dip angle, inva sion, and shoul -der-bed elec tri cal ani so tropy. Anal y sis is car ried fur ther to exam ine the sen si tiv ity of the multi-com po nent mea sure -ments to the extent of the inva sion zone in a devi ated well

and in the pres ence of shoul der-bed ani so tropy. Finally,we exam ine the response due to non-sym met ric inva sionin a hor i zon tal well by con sid er ing the inter play of sev eral petrophysical prop er ties involved in the pro cess ofmud-fil trate inva sion and their effect on multi-com po nent induc tion mea sure ments.

Sim u la tions show that shoul der-bed effects across sand lay ers can be sig nif i cant in the pres ence of shoul der-bedani so tropy, even at low val ues of dip angle. Mea sure -ments cen tered about sand lay ers exhibit sen si tiv ity to thedepth of mud-fil trate inva sion. It is found that trans -verse-coil mea sure ments exhibit dif fer ent responses to the inva sion fronts, indi cat ing the poten tial abil ity ofmulti-com po nent induc tion mea sure ments to detectnon-uni form inva sion. In addi tion, shoul der-bed ani so -tropy has a con sid er able effect on inva sion sen si tiv i ties tosig nif i cantly alter the assess ment of inva sion in terms ofinvaded- and vir gin-zone resistivities, radial length, andfront shape.

Keywords: Induc tion logs, numer i cal sim u la tion,high-angle shoul der bed effects, inva sion mod el ing

Manu script received by the Edi tor August 8, 2006; revised manu script received March 29, 2007.1Orig i nally pre sented at the 47th SPWLA Annual Log ging Sym po sium, Veracruz, Mex ico, June 4-7, 2006, Paper PPP.2The Uni ver sity of Texas at Aus tin, Depart ment of Petro leum and Geosystems Engi neer ing, One Uni ver sity Sta tion, Mail Stop C0300,Aus tin, Texas, 78712; E-mail: [email protected], [email protected].©2007 Soci ety of Petrophysicists and Well Log Ana lysts. All rights reserved.

pro vide addi tional mea sure ments nec es sary to accu ratelyquan tify anisotropic near-wellbore for ma tion prop er ties.

There are numer ous stud ies on the use of axial arrayinduc tion mea sure ments in com plex bore hole envi ron -ments (Alpak et al., 2003, Ander son et al., 1999, 1997).How ever, less infor ma tion is avail able that dis cusses geo -met ri cal, envi ron men tal, and petrophysical effects onmulti-com po nent EM bore hole mea sure ments. Moran andGianzero (1979), Kriegshaüser et al. (2000), Homan et al.(2003), Rosthal et al. (2003), Bar ber et al. (2004),Tompkins et al. (2004) and Xue et al. (2006) ana lyzed bore -hole and inva sion effects on multi-com po nent induc tionmea sure ments in a ver ti cal well. Davydycheva et al. (2003)and Wang and Fang (2001) inves ti gated multi-com po nentinduc tion responses in a devi ated well with inva sion, andWang et al. (2003) exam ined the effects of bore hole prop er -ties (mud con duc tiv ity and tool eccen tric ity) and inva siondepth within an infi nitely thick layer.

This paper con sid ers the effect of sev eral geo met ri cal,envi ron men tal, and petrophysical param e ters onmulti-com po nent induc tion mea sure ments acquired inhigh-angle and hor i zon tal wells. Spe cial empha sis is givento the pres ence of elec tri cal ani so tropy in the con duc tiveshale lay ers shoul der ing a resis tive, iso tro pic, hydro car -bon-bear ing sand (Clavaud and LaVigne, 2003), and itssub se quent influ ence on mea sure ments acquired across thesand layer under vary ing param e ters, includ ing dip angle,layer thick ness, and inva sion. A suite of sim u la tions is per -formed for vari a tions of the model shown in Fig ure 1 toexam ine the rel a tive effects of bore hole, shoul der-bed ani -so tropy, and inva sion on the dif fer ent com po nents. Sub se -quently, we per form a sys tem atic study of the sen si tiv ity of

multi-com po nent induc tion mea sure ments to radial lengthof inva sion at var i ous dip angles in the pres ence of shoul -der-bed ani so tropy. In addi tion to para met ric mod els, weexam ine mod els gen er ated directly from the sim u la tion ofmud-fil trate inva sion to assess the effect of somepetrophysical prop er ties on the spa tial dis tri bu tion of elec -tri cal resis tiv ity around the bore hole, and ulti mately, onmulti-com po nent induc tion mea sure ments. This pro videsus with accu rate rep re sen ta tions of non-sym met ric andnon-uni form inva sion as a result of well devi a tion, per me -abil ity ani so tropy, cap il lary pres sure, grav ity seg re ga tion,and high/low per me able layer inter faces. Our objec tive is to quan tify the rel a tive sen si tiv ity of multi-com po nent mea -sure ments to sev eral geo met ri cal, envi ron men tal, andpetrophysical param e ters in an effort to eval u ate their abil -ity to dis tin guish the var i ous effects and thereby obtain anaccu rate and reli able char ac ter iza tion of for ma tion prop er -ties.

Cal cu la tions are car ried out with a finite-dif fer ence codebased on a cou pled vec tor-sca lar poten tial for mu la tion(Hou et al., 2006). This code per forms accu rate sim u la tionsof multi-com po nent bore hole induc tion mea sure ments in3D inhomogeneous, anisotropic media over a wide range of fre quen cies. Benchmarking of the code indi cates that theaccu racy of the sim u la tions con sid ered in this paper iswithin 2%.

278 PETROPHYSICS August 2007

Mallan and Torres-Verdín

FIG. 1 Lay ered resis tiv ity model with bore hole and inva sion.Diam e ters of the bore hole and inva sion are 21.59 cm and 82.55cm, respec tively. The bore hole resis tiv ity is 0.4 ohm-m, with thebore hole dip angle located in the x-z Car te sian plane.

FIG. 2 Out crop of a turbidite sed i men tary sequence empha siz -ing the elec tri cal anisotropic nature of shale units and the elec tri -cal iso tro pic nature of sand units (pho to graph cour tesy of ArturoContreras).

METHODOLOGY

Bore hole EM induc tion log ging mea sure ments are sim u -lated across a com plex res er voir model with per me able andimper me able lay ers. Fig ure 1 shows the resis tiv ity modelcon sid ered for the numer i cal sim u la tions. It con sists of asequence of res er voir lay ers, rang ing from con duc tivebrine-bear ing to resis tive hydro car bon-bear ing sandsshoul dered by trans versely iso tro pic (TI) con duc tive shalelay ers. This model was adapted from actual log mea sure -ments acquired in a deep water Gulf of Mex ico well pen e -trat ing Ter tiary uncon sol i dated turbidite sed i ments. Itincludes a wide range of vari a tions of con duc tiv ity/resis tiv -ity con trasts. Fig ure 2 shows an out crop ana logue of theturbidite clastic sed i men tary sequence cap tured by themodel described in Fig ure 1. The out crop ana logue sug -gests the TI elec tri cal char ac ter of shale units as well as theelec tri cal iso tropy of sand units.

We sim u late the mea sure ments assum ing a tool that con -sists of col lo cated multi-com po nent mag netic receiv ersposi tioned 1.8 and 0.4 m below a sin gle col lo catedmulti-com po nent mag netic trans mit ter oper at ing at 25 and100 kHz (Fig ure 3). The trans mit ter-receiver off sets werecho sen to exam ine the rel a tive responses for short and longreceiver spac ings. These off sets are con sis tent with exist ing com mer cial tech nol o gies (Bar ber et al., 2004).

The numer i cal sim u la tion algo rithm over lays an opti -mized grid onto a parameterized model domain and assignsaver aged ten sor con duc tiv i ties to each cell. A Car te sian grid is con structed around the tool with a min i mum grid size of0.05 m near the trans mit ter and receiv ers (Fig ure 3). Gridsteps pro gres sively increase out ward from trans mit -ter/receiver loca tions with incre men tal steps as describedby Davydycheva et al. (2003), to the outer bound ary located 1–2 skin depths away for the low est fre quency of inter est. A homo ge neous zero Dirichlet bound ary con di tion isenforced at the outer grid nodes. The for ma tion modelshown in Fig ure 1 is super im posed onto the grid using thestan dard con duc tiv ity aver ag ing approach described byMoskow et al. (1999). In the sim u la tion exam ples, thefinite-dif fer ence grid con sists of 28 x 28 x 92 cells in the x,y, and z direc tions, respec tively. Fig ure 3 shows the dis tri -bu tion of grid nodes in the x-z Car te sian plane super im -posed to one of the lay ers included in Fig ure 1, dip ping at an angle of q. The same fig ure shows the rel a tive loca tion ofthe trans mit ter and receiv ers. We shift the grid from onemea sure ment point to another as the loca tion of the tool ispro gres sively moved along the bore hole axis to repro duce a stan dard log ging oper a tion. The finite-dif fer ence mod el ingcode sim u lates a trans mit ter as a square loop of cur rent ele -ments located along the edges of the cells imme di ately adja -cent to trans mit ter loca tions. In the sim u la tions, we assume

that the loop radius is 5 cm, and that the trans mit ter momentis one. We note that the sim u lated responses of sin gle-loopcoils may not in gen eral accu rately rep re sent responsesfrom finite-length sole noid coils at the short 0.4 m spac ing.How ever, the work by Kriegshaüser et al. (2000) shows that multi-com po nent mag netic fields due to finite-length sole -noids can be approx i mated with those due to sin gle coilseven at the rel a tively short dis tance of 0.4 m within the fre -quency range con sid ered in this paper.

Sim u lated mag netic fields for the direct-cou pled com po -nents are con verted, or scaled, to appar ent con duc tiv i tiesusing the rela tion ships (Zhdanov et al., 2001)

s pp

pp

pp

H

K=-

Á, (1)

where

KL

zz =wm

p

0

4, (2)

K KL

xx yy= =wm

p

0

8, (3)

and w is the angu lar fre quency, m0 is the free-space mag netic per me abil ity, L is the trans mit ter-receiver sep a ra tion, andÁHpq is the quad ra ture com po nent of the mag netic fieldmea sured by the receiver ori ented in the q direc tion andexcited by the trans mit ter ori ented in the p direc tion. These


Effects of Geo met ri cal, Envi ron men tal, and Petrophysical Param e ters on Multi-Com po nent Induc tion Mea sure ments Acquired in High-Angle Wells

FIG. 3 Sub set of the 28 x 28 x 92 finite-dif fer ence grid used forthe numer i cal sim u la tions. Over lain on the grid are the ver ti calbore hole and the first layer, with inva sion, dip ping at q. To theright of the grid is a depic tion of the tool con fig u ra tion, roughly toscale and ver ti cally aligned with the grid. Note the align ment ofthe direc tional com po nents of the grid and the tool.

Kpp scal ing fac tors are con sis tent with those listed by Wang

et al., (2006) that relate appar ent con duc tiv ity to receiver

volt age and trans mit ter cur rent. Fur ther more, Wang et al.,

(2006) define scal ing fac tors for the cross-cou pled com po -

nents, which we adopt here in con sis tent fash ion as

K K K KL

zx xz zy yz= = = =wm

p

0

16, (4)

and

K KL

xy yx= =wm

p

0

8. (5)

NUMERICAL RESULTS

Log ging sim u la tions

Multi-com po nent log ging responses are sim u lated forthe model shown in Fig ure 1, for 30°, 60°, and 80° dip, and0° strike. To eval u ate bore hole and inva sion effects, weshow cases for (a) a 1D lay ered model, (b) a lay ered modelwith bore hole, and (c) a lay ered model with bore hole andinva sion. In each case, sim u la tions are per formed for thelay ered model hav ing either iso tro pic shale lay ers (shoul der beds) or trans versely anisotropic shale lay ers, where thever ti cal con duc tiv ity is reduced by a fac tor of 10(sh /sv = 10). Results are pre sented for the direct-cou pled(xx, yy, zz) and the cross-cou pled (zx, xz) com po nents. They-directed cross-cou pled com po nents (zy, yz, xy, yx) arezero as a result of the assumed model sym me try in they-direc tion (tool azi muth is zero). Fig ure 4, Fig ure 5, and

Fig ure 6 show model responses at 30°, 60°, and 80° dip,respec tively, for a tool with 0.4 m trans mit ter-receiver off -set and oper at ing at 25 kHz. Sim i larly, Fig ure 7, Fig ure 8,and Fig ure 9 show model responses at 30°, 60°, and 80° dip, respec tively, for a tool with 1.8 m trans mit ter-receiver off -set and oper at ing at 25 kHz. In these fig ures each panelincludes the lay ered con duc tiv ity model that shows thedecreased ver ti cal con duc tiv ity in the shale lay ers whenani so tropy is included in the model. For the mod eledresponses, solid lines iden tify cases with iso tro pic shalelay ers, and dashed lines (of cor re spond ing color) iden tifycases with anisotropic shale lay ers. In the sub sec tionsbelow, we dis cuss the log ging sim u la tions in order ofincreas ing model com plex ity for the cases with iso tro picshale lay ers. We first dis cuss the 1D lay ered modelresponses as a func tion of dip angle fol lowed by the sub se -quent rel a tive effects due to bore hole and inva sion. We next dis cuss how the lat ter effects are impacted by the pres enceof ani so tropy in the shale lay ers.

Bed bound ary effect

Sim u la tions for the 1D lay ered model with iso tro picshale lay ers indi cate that layer bound aries are less definedby the lon ger 1.8 m off set responses, as expected, due to thelarger vol ume of the model across which the response isaver aged. The char ac ter is tic “horns” in the 0.4 m sxx and syy

responses are caused by the accu mu la tion of elec tri calcharge at bed bound aries. This charge buildup is asso ci atedwith elec tri cal cur rent flow ing nor mal to the layer bound -ary. With increas ing dip angle, the Hxx-induced cur rentflows increas ingly par al lel to the bed bound ary and, con -versely, the Hzz-cur rent flow becomes increas ingly nor mal.



FIG. 4 (a) szz, (b) sxx, (c) syy, (d) szx, and (e) sxz nor mal ized responses for vari ants of the model shown in Fig ure 1. Receiver off set is0.4 m and oper at ing fre quency is 25 kHz. The dip angle is 30°, mea sured from the nor mal direc tion to the lay ers. Pan els show sim u la -tions for the model with iso tro pic (solid lines) and anisotropic (dashed lines) shoul der beds.

a b ec d

As a result, the horns observed in the sxx response dimin ishwith increas ing dip angle. At the same time, devel op ment of horns occurs in the szz response. The horns for syy remain inthe sim u la tions because the ori en ta tion of the induced cur -rent loop with respect to the layer bound ary does not change with dip. We empha size the sym me try in the szz, sxx, and syy

responses with respect to the responses at the top and bot -tom of each sand layer. On the other hand, the szx and sxz

responses are more asym met ric in nature. This behav ior is aresult of the spa tial region of sen si tiv ity of the respec tivecom po nents, where the zz, xx, and yy com po nents are asso -

ci ated with sym met ric spa tial sen si tiv ity func tions, and thezx and xz com po nents are asso ci ated with non-sym met ricspa tial sen si tiv ity func tions (Alumbaugh and Wilt, 2001).Fur ther more, the excur sions in szx and sxz at bed bound ariesare shifted below and above the layer bound aries, respec -tively, due to the non-sym met ric nature of their respec tivespa tial sen si tiv ity func tions, where, due to rec i proc ity, theszx spa tial sen si tiv ity func tion is enhanced around thez-trans mit ter, and the sxz spa tial sen si tiv ity func tion isenhanced around the z-receiver.




a b dc e


a b dc e

Bore hole effect

In gen eral, sim u la tions for the lay ered model with bore -hole with iso tro pic shale lay ers indi cate non-neg li gi blebore hole effects that include a uni form shift in the 0.4 moff set szz, sxx, and syy responses, where the direc tion of theshift across the more con duc tive 0.3 ohm-m sand layer isoppo site to that in the less con duc tive sand lay ers. Thisbehav ior is observed when the bore hole con duc tiv ity isgreater or less than the neigh bor ing for ma tion con duc tiv ity(Wang et al., 2003). Fur ther more, the shift direc tion in szz isoppo site to that in sxx and syy, as observed by Ander son et

al. (2002). In con trast to the uni form shift, the bore holeeffect toward sxx and syy at the bed bound aries is irreg u lar,espe cially across the more con duc tive 0.3 ohm-m sandlayer where at 60° dip the sxx horns are com pletelysmoothed. The bore hole effect on the lon ger 1.8 m off setresponses is almost neg li gi ble. The cor re spond ing bore hole effect on the cross-cou pled szx and sxz responses is todecrease the size of the excur sions at bed bound aries.

Inva sion effect

Sim u la tions for the lay ered model in the pres ence of




a b dc e


a b dc e

bore hole, inva sion and iso tro pic shale lay ers empha size thecon tri bu tion of inva sion effects on mea sure ments acquiredwith the 0.4 m off set. These mea sure ments include the szz

response across all sand lay ers and the sxx and syy responsesacross the con duc tive 0.3 ohm-m layer. The sxx and syy

responses across the more resis tive 10 ohm-m and 80ohm-m lay ers appear unaf fected and there fore insen si tiveto inva sion. This is an incor rect assess ment because, as weshow in the fol low ing sec tion, inva sion effects on the sxx

and syy responses are at a point of cross ing from a decreased response to an increased response, spe cific to the case of aradial length of inva sion of 0.3 m in the sim u lated mod els.The con duc tiv ity of the invaded zone in each sand layerdecreases the con duc tiv ity con trast with the shoul der ingshale lay ers, thereby caus ing the horns in the sxx and syy

responses at the bed bound aries to be reduced, if not com -pletely smoothed. Inva sion effects on the lon ger 1.8 m off -set responses are rel a tively small, but vis i ble in the sxx andsyy responses across the 80 ohm-m and 10 ohm-m lay ers,becom ing more appar ent across the 0.3 ohm-m layer. Thesxx response is only slightly affected at the large 60° and 80° dips across the 0.3 ohm-m layer and near the bed bound -aries of the 80 ohm-m and 10 ohm-m lay ers. The effect onthe cross-cou pled szx and sxz responses is to decrease thesize of the excur sions at bed bound aries, where the cor re -spond ing effect is more pro nounced in the short 0.4 m off -set.

Shoul der-bed ani so tropy effect

Sim u la tions for the lay ered mod els dis cussed above, butwith anisotropic shale lay ers, show that multi-com po nentresponses across the sand lay ers are affected in dif fer ent

ways by the pres ence of ani so tropy in the shoul der ing shalelay ers. The cor re spond ing effect on the szz response is min i -mal at low dip, but becomes more prom i nent with increas -ing dip, and causes a decreased szz response. Con versely,the sxx response has a prom i nent neg a tive shift at low dip,and dimin ishes with increas ing dip such that the effect ismin i mal at 80° dip. The reverse effect between the szz andsxx responses is a result of the ori en ta tion of the inducedcur rents with respect to the direc tion of the trans verse ani -so tropy. With increas ing dip angle, the Hzz-induced cur rentflows increas ingly nor mal to the sh plane and, con versely,the Hxx-cur rent flow becomes increas ingly par al lel. Theeffect on the syy response, where the ori en ta tion of theHyy-induced cur rent with respect to the ani so tropy isunchanged with dip, tran si tions from a neg a tive shift at lowdip to a pos i tive shift at high dip, such that the effect on the1.8 m off set response at 60° dip is almost neg li gi ble. Thisbehav ior is explained by the dis tor tion of the yy spa tial sen -si tiv ity func tion caused by ani so tropy, as dis cussed byTompkins et al. (2004), where the tool axis and the spa tialsen si tiv ity func tion become mis aligned with increas ing dip. The receiver tran si tions from a neg a tive to a pos i tive spa tial sen si tiv ity region. Fun da men tally, the effect of ani so tropyis to flat ten the Hyy-induced cur rent ‘loop’, thereby caus ingthe cur rent loop to become ellip ti cal and elon gated in the sh

plane (Weiss et al., 2001). As a con se quence, the prox im ityof the y-receiver to the induced cur rent changes with dip.

Numer i cal sim u la tions gen er ally indi cate that the effectof shoul der-bed ani so tropy on the mea sure ments is morepro nounced for the con duc tive 0.3 ohm-m res er voir layer,com pared to a much smaller effect across the more resis tive80 ohm-m and 10 ohm-m lay ers. These results reflect the




a b dc e

model with res er voir lay ers that have both dif fer entresistivities and thick nesses, which adds ambi gu ity as towhich param e ter, resis tiv ity or thick ness, influ ences theshoul der-bed ani so tropy effect across the res er voir layer. To shed fur ther light to this prob lem, we repeated the sim u la -tions for the 1D lay ered model, where, for each sim u la tionres er voir lay ers were assigned the same resis tiv ity, 80, 10,or 0.3 ohm-m. The ensu ing sim u la tions show that, for thegiven model, the resis tiv ity, and not the thick ness of thelayer plays a sig nif i cant role on the effect that shoul der-bedani so tropy has on mea sure ments acquired across the res er -voir layer. This is due in part to the rel a tive change in resis -tiv ity con trast between the res er voir and shale lay ers whenani so tropy is included in the model. Addi tion ally, a largercur rent den sity is induced in a more con duc tive res er voirlayer, and dis tor tion of this cur rent flow due to pres ence ofani so tropy in the neigh bor ing shale lay ers has a greaterimpact on the induc tion mea sure ment.

Sen si tiv ity to inva sion

We exam ine the sen si tiv i ties of the dif fer ent induc tioncom po nents to radial length of inva sion in a devi ated wellenvi ron ment. Wang et al. (2001) mod eled the inva sioneffect in devi ated wells for two dif fer ent inva sion depthsinto anisotropic res er voir lay ers. Bar ber et al. (2004)described the multi-com po nent responses to inva sion thick -ness in the form of an inte grated Born response for a ver ti -cal well pen e trat ing an infi nitely thick layer, i.e.whole-space. Here, we model inva sion into a dip ping, iso -tro pic res er voir layer in the pres ence of iso tro pic andanisotropic shoul der beds in order to eval u ate the effect ofshoul der-bed ani so tropy on the inva sion response. Ourobjec tive is to study the rel a tive shape of the responses withrespect to radial length of inva sion; a largely vary ingresponse indi cates greater sen si tiv ity to quan tify the inva -sion front. It is also impor tant to assess whether pres ence ofani so tropy affects the shapes of these responses.

To better quan tify the inva sion response and the sub se -quent effect of shoul der-bed ani so tropy, we per form sim u -la tions for increas ing radial lengths of inva sion. The toolsource-receiver mid point is cen tered in the 10 ohm-m res er -voir layer (Fig ure 1), but with the layer thick ness decreasedto 0.9 m (that of the 0.3 ohm-m res er voir layer) to better illus -trate the effect of neigh bor ing lay ers. Responses are cal cu -lated for increas ing val ues of radial length of inva sion for themodel with iso tro pic and anisotropic shoul der beds. Due tomodel sym me try in the y-direc tion, the y-directed cross-cou -pled com po nents (zy, yz, xy, yx) are zero. Fur ther more, sincethe source-receiver mid point is cen tered in the layer, thecross-cou pled com po nents, zx and xz, are iden ti cal. Thus,only the szz, sxx, syy, and szx responses are stud ied here.

Fig ure 10 shows szz, sxx and syy responses, for 1.8 m off -set and 25 kHz fre quency, at 30°, 60°, and 80° dip. Sim i -larly, Fig ure 11 shows 0.4 m off set results. At the low 30°dip with iso tro pic shoul der beds, the 1.8 m and 0.4 mresponses resem ble the cumu la tive response func tionsdescribed by Bar ber et al. (2004). The gen eral nature ofthese responses are under stood by the whole-spacemulti-com po nent spa tial sen si tiv ity func tions (or sen si tiv -ity vol umes) exam ined by Alumbaugh and Wilt (2001),Tompkins et al. (2004), and Bar ber et al. (2004).Whole-space sen si tiv i ties are most cer tainly altered by thepres ence of shoul der beds, and would dif fer with dip angle.An increase in szz responses is observed as the inva sionfront advances through the pos i tive spa tial sen si tiv ityregion of the zz com po nent, beyond which the responsesbegin to asymp tote. Asymp totic val ues rep re sent val uescom puted for infi nite radial lengths of inva sion and are dis -played with the solid gray lines in the plots. The sxx and syy

responses ini tially decrease to a radial length of inva sion, atwhich the responses increase and even tu ally begin toasymp tote as the inva sion extends beyond the region of spa -tial sen si tiv ity. Such a decrease of the response occurs as the inva sion front advances from the neg a tive to the pos i tivespa tial sen si tiv ity regions of the xx and yy com po nents.Once more these fea tures occur at shal lower and deeperinva sion lengths in the 0.4 m and 1.8 m responses, respec -tively, and illus trate the effect of dif fer ent radial lengths ofinves ti ga tion. The responses change with increas ing dipangle, espe cially between sxx and syy, as the model becomesthree-dimen sional, and the inva sion pro gresses through dif -fer ent regions of the spa tial sen si tiv ity func tions.

Pres ence of ani so tropy in the shoul der beds has a pro -nounced effect toward the multi-com po nent responses.Ani so tropy can sig nif i cantly alter the spa tial sen si tiv ityfunc tions (Tompkins et al., 2004) due to the dis tor tion of the induced cur rent flow paths (Weiss et al., 2001). Com paredto the iso tro pic responses, a uni form decrease is observed in szz at a short radial length of inva sion out to a point ofinflec tion, beyond which the devi a tion between the iso tro -pic and anisotropic responses increases, and the anisotropicresponse begins to asymp tote at a shorter inva sion depth.Regard ing the sxx and syy responses, the effect of shoul -der-bed ani so tropy alters the radial sen si tiv ity in a non lin -ear fash ion and causes the responses to exhibit asymp toticbehav ior at a shorter radial length of inva sion, indi cat ing adecreased radial extent of the sen si tiv ity. Fur ther more,under iso tro pic shoul der-bed con di tions, the rel a tive shapes of the sxx and syy responses can dif fer sig nif i cantly, depend -ing on dip angle. How ever, in the pres ence of shoul der-bedani so tropy the shapes of the sxx and syy responses tend topar al lel each other, with the excep tion that the inflec tionpoint occurs at a shorter dis tance for syy that depends on the





FIG. 10 szz, sxx, and syy responses ver sus radial length of inva sion (Lxo) for the 3D model shown in Fig ure 1, with and with out elec tri -cal ani so tropy in the shale lay ers. Fre quency is 25 kHz, receiver off set is 1.8 m, and the tool is cen tered in the 10 ohm-m layer (layerthick ness equal to 0.9 m). The well dip angle is (a,b) 30°, (c,d) 60°, and (e,f) 80°. Solid gray lines describe the cor re spond ingresponses for an infi nite radial length of inva sion.

a b

c d

e f



FIG. 11 szz, sxx, and syy responses ver sus radial length of inva sion (Lxo) for the 3D model shown in Fig ure 1, with and with out elec tri -cal ani so tropy in the shale lay ers. Fre quency is 25 kHz, receiver off set is 0.4 m, and the tool is cen tered in the 10 ohm-m layer (layerthick ness equal to 0.9 m). The well dip angle is (a,b) 30°, (c,d) 60°, and (e,f) 80°. Solid gray lines describe the cor re spond ingresponses for an infi nite radial length of inva sion.

a b

c d

e f

dip angle. Fig ure 12 shows the cor re spond ing 25 kHz,cross-cou pled szx responses for the 0.4 m and 1.8 m off sets.These responses exhibit sen si tiv ity to dip angle and are sig -nif i cantly altered in the pres ence of shoul der-bed ani so -tropy. We note that at 80° dip, the influ ence of shoul der-bedani so tropy is neg li gi ble toward the sxx response, becausethe xx induced cur rents flow almost par al lel to the sh plane.

Responses of sxx and syy at a fre quency of 100 kHz for30°, 60°, and 80° dip exhibit dif fer ent sen si tiv i ties to inva -sion, such as in their asymp totic behav ior, for the iso tro piccase, at 60° and 80° dip angles, and the rel a tive shifts due topres ence of ani so tropy (Fig ure 13). These dif fer ences indi -cate addi tional, unique infor ma tion born by the fre quencybehav ior of the var i ous com po nents.

Hor i zon tal well

Alpak et al. (2003) exam ined the sen si tiv ity of an axialarray induc tion response to numer i cally sim u lated mud-fil -trate inva sion from a hor i zon tal well into a thick layer withvary ing degrees of per me abil ity ani so tropy. Using Archie’sequa tion, they con verted sim u lated spa tial dis tri bu tions offluid sat u ra tion and fluid resis tiv ity to spa tial dis tri bu tionsof bulk iso tro pic resis tiv ity, for which induc tion responseswere cal cu lated. Here, we exam ine the multi-com po nentsen si tiv i ties to the non-uni form mud-fil trate inva sion in ahor i zon tal well using sim u la tion results from Mendoza etal. (2007). The lat ter authors sim u lated water-base mud-fil -

trate inva sion from a hor i zon tal well into a gas-bear ing,

infi nitely thick sand layer. Bore hole diam e ter is 20.32 cm,

and the bore hole mud has a salt con cen tra tion of 5,000 ppm.

The for ma tion poros ity is 25%, per me abil ity is 300 mD, ini -

tial water sat u ra tion is 0.3, and con nate water salt con cen -

tra tion is 150,000 ppm.The 2D grids of the sim u lated spa tial dis tri bu tions of

water sat u ra tion and salt con cen tra tion are con verted, cell

by cell, to sca lar (or iso tro pic) con duc tiv i ties using Archie’s

equa tion (Archie, 1942)

s s f= ( / ) ,1 a Swm

wn (6)

where s, sw, f, and Sw denote the for ma tion con duc tiv ity,brine con duc tiv ity, poros ity, and brine sat u ra tion, respec -tively, with the empir i cal con stants a = 1, m = 2, and n = 2.Brine con duc tiv ity is cal cu lated with the equa tion (Zhang etal., 1999)

s w

wC T= +

æ

èç

ö

ø÷

+

é

ëê

ù

ûú

-

0.01233647.5

0 995

182

18 39. ., (7)

where Cw is the sim u lated salt water con cen tra tion (in ppm),and T is for ma tion tem per a ture (in °C), assumed here equalto 98.9 °C. Fig ure 14 shows cross-sec tions per pen dic u lar tothe bore hole axis of the cal cu lated con duc tiv ity dis tri bu -tions for dif fer ent cases of per me abil ity ani so tropy, wherethe bore hole mud con duc tiv ity is 2.4 S/m, and the back -



FIG. 12 Cross-cou pled szx response ver sus radial length of inva sion (Lxo) for the 3D model shown in Fig ure 1, with and with out elec -tri cal ani so tropy in the shale lay ers. Fre quency is 25 kHz, receiver off sets are (a) 0.4 m and (b) 1.8 m, and the tool is cen tered in the 10 ohm m layer (layer thick ness equal to 0.9 m). The well dip angle is 30°, 60°, and 80°. Solid gray lines describe the cor re spond ingresponses for an infi nite radial length of inva sion.

a b

ground for ma tion con duc tiv ity is 0.28 S/m. The con duc tiveannu lus dis played in the cross-sec tions is a result of the“dephasing” of the water con cen tra tion front with respect tothe salt con cen tra tion front (George et al., 2004). This sca lar con duc tiv ity grid is mapped, using bilinear inter po la tion, toa 44 × 44 xy-grid with an outer bound ary sim i lar to that usedin the pre vi ous EM log ging sim u la tions. The xy-plane con -duc tiv i ties are cop ied in the z-direc tion of the grid, therebygen er at ing a 3D mesh to accom mo date the 3D EMfinite-dif fer ence mod el ing code. Tool ori en ta tion is suchthat the direc tional com po nents remain aligned with thegrid, where the x and z com po nents are in the hor i zon talplane, and the y com po nent is in the ver ti cal direc tion.

Multi-com po nent induc tion responses are cal cu lated formul ti ple receiver off sets, rang ing from 0.4 m to 1.8 m, at afre quency of 100 kHz. These responses are con verted toappar ent con duc tiv i ties using equa tion (1), where Kpp isobtained numer i cally and relates the sim u lated whole-space response to the back ground con duc tiv ity of 0.28 S/m asso -ci ated with the sim u la tions of mud-fil trate inva sion. Thecal i bra tion con stant for the cross-cou pled com po nent Kzy = 1

4 Kzz to be con sis tent with equa tion (4).Fig ure 15 describes szz, sxx, syy, and szy responses for the

mod els shown in Fig ure 14. Due to model sym me try in thex-direc tion, the x-con tain ing cross-com po nent responsesare zero, and because the model does not change in thez-direc tion, it fol lows that szy = syz. Responses are includedfor a whole-space, to ref er ence true for ma tion con duc tiv ity,



FIG. 13 sxx and syy responses ver sus radial length of inva sion(Lxo) for the 3D model shown in Fig ure 1, with and with out elec -tri cal ani so tropy in the shale lay ers. Receiver off set is 1.8 m andthe tool is cen tered in the 10 ohm-m layer (layer thick ness equalto 0.9 m). The well dip angle is (a) 30°, (b) 60°, and (c) 80° , andthe oper at ing fre quency is 100 kHz. Solid gray lines describe the cor re spond ing responses for an infi nite radial length of inva sion.

a

c

b

FIG. 14 Spa tial dis tri bu tions of elec tri cal con duc tiv ity cal cu -lated from the hor i zon tal mud-fil trate sim u la tions described byMendoza et al. (2007). Pan els (a), (b), (c), and (d) show casesfor per me abil ity ani so tropy with kh /kv = 1, 3, 5, and 10, respec -tively. Note that the direc tional com po nents of the tool arealigned with the sim u la tion grid.

and for a well with no inva sion to ref er ence bore hole effect.

For all mod els, inva sion responses are dis tin guish able from

those asso ci ated with no inva sion. With excep tion of the

kh /kv = 1 case, the szz and syy inva sion responses dif fer neg -

li gi bly among the cases. The sxx and szy responses, how ever,

are dis tin guish able among the dif fer ent mod els. In pan els

(b), (c), and (d), of Fig ure 14, the change of con duc tive

annu lus occurs mostly azimuthally, rather than radi ally,

around the bore hole, which explains the mar ginal vari a tion

of the szz responses. The change is more sig nif i cant in the

x-direc tion when con sid er ing slices that transect the bore -

hole. This explains the rel a tively larger vari a tions in sxx ver -

sus syy. The non-sym me try about the bore hole of the con -

duc tiv ity dis tri bu tion in the y-direc tion gives rise to thenon-zero cross-cou pled szy response.

The anal y sis is extended for the case of per me abil ity ani -so tropy with kh /kv = 10 to super im pose the effect of elec tri -cal ani so tropy due to small shale lami na tions. Anisotropiccon duc tiv i ties are assigned cell by cell, where each cell’sver ti cal con duc tiv ity in the y-direc tion is reduced by a fac -tor of 10 with respect to cell’s hor i zon tal con duc tiv i ties inthe x- and z-direc tions that rep re sent the inter po lated sca larcon duc tiv ity. Fig ure 16 shows results for this sim u la tion.The effect of ani so tropy uni formly shifts the szz and szy

responses to a decreased appar ent con duc tiv ity and shiftsthe syy response to an increased appar ent con duc tiv ity. Fur -ther more, the szz responses (with and with out inva sion)



FIG. 15 Sim u la tions of appar ent con duc tiv ity against receiver off set for the con duc tiv ity dis tri bu tions shown in Fig ure 14. The pan els show (a) szz, (b) sxx, (c) syy, and (d) szy responses. Also included are results for a back ground whole-space and for the well with noinva sion. The oper at ing fre quency is 100 kHz.

a b

c d

become less dis tin guish able, which indi cates that thezz-induced cur rents are dom i nated more by ani so tropyrather than by inva sion. By con trast, the sxx responsechange is non lin ear, with the appar ent con duc tiv ity increas -ing with increas ing receiver off set. The large ani so tropyeffects on the szz and sxx responses arise because theirrespec tive induced cur rents flow in the ver ti cal direc tion. Asmall ani so tropy effect on the syy response exists becausethe dis tor tion of the induced cur rent, caused by the non-uni -form inva sion, causes a por tion of the cur rent to flow ver ti -

cally. This is evi dent by the exis tence of the cross-cou pledszy response. The non lin ear increase in the sxx response, due to ani so tropy, may be explained by the elon ga tion of theHxx-induced cur rent loop along the axis of the tool, as illus -trated by Weiss et al. (2001).

The sim u lated tool responses are repeated, but with thetool eccentered in the bore hole. Eccen tric ity is sim u lated by shift ing the tool, or the xy-grid cen ter, down ward (withrespect to the con duc tiv ity model) in the y-direc tion by 5cm – approx i mately half way to the bore hole wall. These



FIG. 16 (a) szz, (b) sxx, (c) syy, and (d) szy responses against receiver off set for the con duc tiv ity dis tri bu tion in Fig ure 14d (kh /kv = 10).Pan els show responses for the cases with and with out inva sion, and with and with out elec tri cal ani so tropy (sh/sv = 10). For eachcase, the solid line iden ti fies the respec tive response for the tool cen tered in the bore hole, and the dashed line (with cor re spond ingsym bol) iden ti fies the cor re spond ing case for the tool y-eccentered in the bore hole. Also included is the result for a back groundwhole-space. The oper at ing fre quency is 100 kHz.

a b

c d

sim u la tions are shown in Fig ure 16, along side results for

the tool cen tered in the bore hole. In all com po nents, the

eccen tric ity causes a shift in the responses at short receiver

off sets, con sis tent with results described by Davydycheva

et al. (2006), with a more pro nounced shift occur ring in the

sxx response. The eccen tric ity effect dimin ishes for increas -

ing receiver off set. Also, pres ence of ani so tropy appears to

have lit tle effect on the nature of the eccen tric ity effect in

the szz, sxx, and syy responses.The sim u lated tool response is repeated for the same for -

ma tion con duc tiv ity dis tri bu tion, but with the bore hole con -

duc tiv ity decreased to 0.001 S/m to sim u late pres ence of

oil-base mud. Although the con duc tiv ity dis tri bu tions out -

side the bore hole (in Fig ure 11) may not rep re sent sim u -

lated mud-fil trate inva sion of a resis tive oil-based mud, we

nev er the less wish to exam ine the dif fer ence in response

between con duc tive and resis tive bore hole muds for a given

for ma tion con duc tiv ity dis tri bu tion. Refer ring to Fig ure 17,

the rel a tive effect of ani so tropy on the szz, sxx, and szy

responses is sim i lar to that for the con duc tive mud, in that

szz, and szy are uni formly shifted and sxx increases with

receiver off set. By con trast, the syy change is no lon ger par -

al lel, but decreases with increas ing receiver off set.



FIG. 17 (a) szz, (b) sxx, (c) syy, and (d) szy responses ver sus receiver off set for the con duc tiv ity dis tri bu tion in Fig ure 14d (kh /kv = 10),but with the bore hole con duc tiv ity decreased to 0.001 S/m to rep re sent an oil-base mud. Pan els show responses for the cases withand with out inva sion, and with and with out elec tri cal ani so tropy (sh /sv = 10). For each case, the solid line iden ti fies the respec tiveresponse for the iso tro pic for ma tion, and the dashed line (with the cor re spond ing sym bol) iden ti fies the cor re spond ing case for theanisotropic for ma tion. Also included is the result for a back ground whole-space. The oper at ing fre quency is 100 kHz.

a b

c d

CONCLUSIONS

Shoul der-bed ani so tropy has a mea sur able effect oninduc tion log ging responses across a sand layer. Both bore -hole dip angle and con duc tiv ity of the sand layer deter minethe rel a tive con tri bu tion of this effect on the mea sure ments,caus ing pos si ble non-unique ness in dis tin guish ing true for -ma tion prop er ties.

For the radial length of inva sion con sid ered in the model(Fig ure 1), the effect from shoul der-bed ani so tropy is, inmost of the cases pre sented, much more pro nounced thanthe inva sion response. Con se quently, it is impor tant toaccu rately account for shoul der-bed ani so tropy to cor rectlydescribe the sig na ture of inva sion on the mea sure ments.

The sen si tiv ity of the induc tion tool to radial length ofinva sion is also affected by shoul der-bed ani so tropy. Foriso tro pic shoul der beds, the var i ous EM mea sure ment com -po nents exhibit dis tin guish able sen si tiv i ties with respect toeach other, indi cat ing their abil ity to deter mine the shape ofthe inva sion front. We note, how ever, that the shapes ofthese responses can be very sen si tive to dip angle. Con se -quently, an accu rate mea sure ment of the dip angle betweenthe bore hole and the layer is vital to cor rectly assess ingpres ence and shape of inva sion. Shoul der-bed ani so tropyalso causes the sen si tiv ity responses to reach their asymp -totic value at a shorter radial length of inva sion, therebyindi cat ing a decrease in the radial length of sen si tiv ity. Fur -ther more, the rel a tive shapes of the sxx and syy responsesbecome sim i lar, thereby lim it ing the abil ity of these com po -nents to accu rately esti mate the inva sion front.

Sim u la tions for the case of a hor i zon tal well indi cate that mud-fil trate inva sion entails unique responses in the xx andzy com po nents, dem on strat ing the sen si tiv ity of multi-com -po nent induc tion mea sure ments to the shape and spa tialextent of the inva sion front. Pres ence of elec tri cal ani so -tropy causes a uni form shift in the appar ent con duc tiv i tiesesti mated from Hzz and Hyy. By con trast, the con duc tiv ityesti mated from Hxx has a dis tin guish able nature, in that itincreases with increas ing receiver off set. This fea ture ofthe mea sure ments could be used to quan tify pres ence ofelec tri cal ani so tropy in a hor i zon tal well envi ron ment.

The study pre sented in this paper is a pre am ble to inver -sion. It is dif fi cult to intu itively com pre hend the wealth ofinde pend ent and cor re lated infor ma tion included inmulti-com po nent induc tion mea sure ments. How ever, anunder stand ing of the nature and sen si tiv i ties of these mea -sure ments to var i ous geo met ri cal, envi ron men tal, andpetrophysical effects may be incor po rated into inver sionalgo rithms to reduce non-unique ness. For instance, sub setsof the mea sure ments hav ing greater or less sen si tiv ity tospe cific envi ron men tal and geo met ri cal effects may be used in a data-adap tive inver sion scheme, such as dis cussed by

Alpak et al. (2005). In so doing, 1D inver sion for lay ered

struc ture may pre cede 3D inver sion incor po rat ing bore hole

and inva sion. Fur ther more, a rudi men tary assess ment of

for ma tion prop er ties, such as layer ori en ta tion, may be

inter preted from the raw mea sure ments and used as the

start ing model, i.e. ini tial guess, of an iter a tive non lin ear

inver sion pro cess.

NOMENCLATURE

a Parameter in Archie’s equa tionm Poros ity expo nent in Archie’s equa tionn Sat u ra tion expo nent in Archie’s equa tionCw Sim u lated salt water con cen tra tion (ppm)ÁHpq Quad ra ture com po nent of the mag netic field

excited by a trans mit ter ori ented in the p direc-tion and mea sured by a receiver ori ented in q direc tion.

Kpp Scal ing fac tors for direct-cou pled andcross-cou pled com po nents

L Trans mit ter – Receiver sep a ra tion dis tance (m)Sw Water sat u ra tionT For ma tion tem per a ture (°C)f Poros ityspp Appar ent con duc tiv i ties for direct-cou pled

com po nentss For ma tion con duc tiv ity (S/m)sw Brine con duc tiv ity (S/m)q Bore hole dip angle (degrees)w Angu lar fre quencym0 Free-space mag netic per me abil ity

ACKNOWLEDGMENTS

We express our deep est grat i tude to (Ian) Zhiyi Zhang,

Dean Homan, and (Cynthia) Dong Xue for their con struc -

tive tech ni cal and edi to rial feed back. Fund ing for the work

reported in this paper was pro vided by UT Aus tin’s

Research Con sor tium on For ma tion Eval u a tion, jointly

spon sored by Anadarko, Aramco, Baker Atlas, BP, Brit ish

Gas, ConocoPhillips, Chev ron, ENI E&P, ExxonMobil,

Halliburton Energy Ser vices, Mar a thon, Mex i can Insti tute

for Petro leum, Norsk-Hydro, Occi den tal Petro leum Cor po -

ra tion, Petrobras, Schlumberger, Shell Inter na tional E&P,

Statoil, Total, and Weatherford.

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ABOUT THE AUTHORS

Rob ert K. Mal lan received a BSc degree in Phys ics from EastCen tral Uni ver sity in Oklahoma in 1990, and an MSc degree inGeo phys i cal Engi neer ing from the Uni ver sity of Ari zona, Tuc son,in 1995. Dur ing 1996-2003, he worked for Elec tro Mag neticInstru ments (Rich mond, CA) on the devel op ment and appli ca tionof bore hole EM induc tion log ging and magnetotelluric tech nol o -gies. He is cur rently pur su ing a PhD at the Depart ment of Petro -leum and Geosystems Engi neer ing of The Uni ver sity of Texas atAus tin. His research is focused on petrophysics based mod el ingand inver sion of elec tro mag netic and acous tic well log ging data indip ping and anisotropic for ma tions.

Carlos Torres-Verdín received a PhD degree in Engi neer ingGeoscience from the Uni ver sity of Cal i for nia, Berke ley, in 1991.Dur ing 1991–1997 he held the posi tion of Research Sci en tist withSchlumberger-Doll Research. From 1997–1999, he was Res er voirSpe cial ist and Tech nol ogy Cham pion with YPF (Bue nos Aires,Argen tina). Since 1999, he has been with the Depart ment of Petro -leum and Geosystems Engi neer ing of The Uni ver sity of Texas atAus tin, where he cur rently holds the posi tion of Asso ci ate Pro fes -sor. He con ducts research on bore hole geo phys ics, for ma tion eval -u a tion, and inte grated res er voir char ac ter iza tion. Torres-Verdínhas served as Guest Edi tor for Radio Sci ence, and is cur rently amem ber of the Edi to rial Board of the Jour nal of Elec tro mag neticWaves and Appli ca tions, and an asso ci ate edi tor for Petrophysics(SPWLA) and the SPE Jour nal. He is co-recip i ent of the 2003,2004 and 2006 Best Paper Award by Petrophysics, and is recip i entof SPLWA’s 2006 Dis tin guished Tech ni cal Achieve ment Award.



Documents

Effect s of Geo met r i cal, Envi ron m en tal , and ... · August 2007 PETROPHYSICS 277 PETROPHYSICS, VOL. 48, NO. 4 (AUGUST 2007); P. 277–294; ... exam ine the sen si t iv ity