NUMERICAL SIMULATION OF MUD-FILTRATE INVASION IN HORIZONTAL WELLS AND SENSITIVITY ANALYSIS OF ARRAY INDUCTION TOOLS

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1 NUMERICAL SIMULATION OF MUD-FILTRATE INVASION IN HORIZONTAL WELLS AND SENSITIVITY ANALYSIS OF ARRAY INDUCTION TOOLS F. O. Alpak, The Univesity f Texas at Austin, E. B. Dussan V., T. Habashy, Schlumbege-Dll Reseach, and C. Tes-Vedín, The Univesity f Texas at Austin ABSTRACT We develp a pcedue t simulate and detect the spatial distibutin f mud-filtate invasin in the nea-ellbe egin f hizntal ells. Examples ae cnsideed f the geneal theme f a atebase mud invading hydcabn-beaing ck fmatins. A cmmecial multiphase fluid fl simulat is used t quantify the sensitivity f mudfiltate invasin t a lage ange f the ck's petphysical paametes. Spatial distibutins f fluid satuatins endeed by the simulat ae then cnveted int cespnding distibutins f electical cnductivity. Subsequently, a finitediffeence cde is used t simulate the assciated electmagnetic espnse f an aay inductin tl. Seveal imptant cnclusins can be dan fm u sensitivity study: In many pactical cases, the ck's petphysical paametes can have a stng impint n the spatial distibutin f fluid satuatins, and in cnsequence, n the espnse f aay inductin tls. On the the hand, the espnse f aay-inductin tls may exhibit limited sensitivity t seveal f the cmplex gemetical featues f the satuatin distibutin aund a hizntal behle. We have als fund that simple paametic epesentatins f the invasin pcess can be used t intepet quantitatively aay inductin measuements acquied in hizntal ells. Examples ae shn f h aay inductin measuements culd be used t pvide accuate estimates f electic esistivity in the uninvaded egin f ck fmatins. INTRODUCTION Detemining the hydcabn cntent f pus ck fmatins is f significant inteest f the ecnmic assessment f esevis. Accuate deteminatin f in-situ fmatin esistivity as a diect measue f initial il satuatin is ne f the pimay mtivatins f ieline lgging. In additin t assessing eseves, it is impeative t detemine the amunt f ecveable il. Mudfiltate invasin can be egaded analgus t a atefld. Cnsequently, the invaded zne esistivity, R emains a diect indicat f x esidual il satuatin, S. Bth the satuatin pfile and the invasin gemety aund a behle ae gvened by the undelying physics f multiphase fluid fl (Ramakishnan and Wilkinsn, 1997 and 1999). Fm the lg measuement viepint, the pesence f an invaded egin aund the behle can substantially affect the electmagnetic lgs. Depending n the extent f invasin, and n the cmplexity f the invasin gemety, intepetatin f electmagnetic lgs can be nntivial. Undubtedly, the intepetatin f esistivity tls in hizntal highly deviated ells is much me challenging than in vetical ells. Thee is n ideal lgging tl that measues cntinuus esistivity pfiles. Heve, tls ae available ith multiple depths f investigatin. The Aay Inductin Tl (AIT*) pvides 28 a measuements (nt all independent) that can be educed int five eslutin-matched channels. The espnse f multiaay inductin tls has been peviusly analyzed by Andesn et al. (1999) f the cases f highly dipping fmatins and abitay 3D gemeties in the pesence f mudfiltate invasin. Heve, the invasin gemeties cnsideed by that study ee nt explicitly deived fm petphysical and multiphase fl ppeties. Simulatin f the nea-ellbe dynamics using multiphase fl equatins has been peviusly applied t the intepetatin f inductin lgging by Zhang et al. (1999) and Ramakishnan and Wilkinsn, (1997 and 1999) f vetical ell gemeties. The mtivatin f this pape is t quantify the sensitivity f aay inductin tls t the spatial distibutins f nea-ellbe fluids aising fm mud-filtate invasin. Me specifically, e pefm a systematic study t appaise the sensitivity f the AIT* t vaiatins f multiphase *Mak f Schlumbege 1

2 petphysical and fluid ppeties in the pesence f hizntal ells. METHODOLOGY The fist step f u study is t descibe the pemeable medium and satuating fluids in tems f standad petphysical paametes. Spatial distibutins f fluid satuatins ae deived fm the physics f mud-filtate invasin psed in a thee-dimensinal (3D) Catesian cdinate fame. Numeical simulatins f the pcess f mudfiltate invasin ae pefmed using a cmmecial black-il multiphase fluid fl simulat, namely, ECLIPSE 100*. Fluid satuatin distibutins aund a hizntal ellbe ae cmputed at a numbe f ieline lgging times f vaius cmbinatins f petphysical and fluid ppeties. The simulated satuatin fields ae subsequently tansfmed int distibutins f electical esistivity using Achie s la. In each case, e investigate a numbe f spatial smthing appaches that pvide a tansfmatin f the detailed satuatin field int an equivalent smthed spatial distibutin f esistivity. This smthing pcedue is designed t peseve the mst dminant featues f the satuatin field btained fm fine-gid multiphase fl simulatins; it is als designed t descibe bth the magnitude and gemety f the satuatin field in tems f a small numbe f paametes in the esistivity dmain. Vaius degees f spatial smthing ae investigated f each simulatin case. The k pesented hee becmes exceedingly valuable in the seach f an ecnmic pcedue t diectly infe petphysical paametes fm aay inductin data. Finally, AIT* measuements ae simulated using a finitediffeence spectal Lanczs decmpsitin methd, SLDMINV (Duskin et al., 1999). Figue 1 is a flchat that summaizes the steps cnsideed in u simulatin and sensitivity study. SIMULATION OF MUD-FILTRATE INVASION We specifically cnside the case f ate-base mud-filtate invading a hydcabn-beaing fmatin. Mud-filtate invasin is mdeled as the isthemal simultaneus fl f ate and il phases (t-phase fl) in a pemeable medium. Simulatin f isthemal multiphase fluid-fl in pemeable media equies f mass balance and tanspt equatins as ell as a cnstitutive equatin f state. We disegad the pesence f chemical eactins, ck/fluid mass tansfe, and diffusive/dispesive tanspt. The mass balance equatin f the i-th fluid phase (eithe ate il) in 3D spatial cdinates is given by ( ρ φ S ) i t i v + ( ρ υ ) = q, i = 1, K, k, i i vi (1) hee ρ is fluid density, υ is fluid velcity, φ is psity, q v is a suce-sink tem (the diving suce tem), S is phase satuatin, and i, k ae phase index and ttal numbe f phases, espectively. On the the hand, the tanspt equatin is Dacy s la, given by v ki v v υ i = k ( pi γ i Dz ), (2) µ i v hee k is the abslute pemeability tens f the pus medium, k is the elative pemeability, µ is the phase viscsity, p is the phase pessue, γ is the phase specific gavity, and D z is the vetical lcatin bel sme efeence level. Finally, the cnstituent equatin flls fm the equatin f state. The assumptin is als made that bth fluid and ck cmpessibilities ae cnstant ve the pessue ange f inteest, given by and c c fl 1 V = V p 1 φ = φ p T. T = 1 ρ ρ p T, (3) (4) The fmatin is initially fully satuated ith liquid hydcabns at ieducible ate satuatin. We futhe assume that thughut the mud-filtate invasin pcess all the available gas emains disslved in the leic phase; in the ds, the leic phase pessue stays abve bubble pint bth spatially and tempally duing invasin. The flling set f t-phase fl equatins is slved in space and time t simulate mud-filtate invasin (Aziz and Settai, 1979): φ S t B and M k ( p γ D ) = 0, z (5) 2

3 φ S t B M k subject t the satuatin identity hee and ( p P γ D ) c z sc R =, (6) S S = 1, (7) + M ( S M ( S k( S) ρ g ) =, γ =, (8) µ B g k( S) ρ g ) =, γ =. (9) µ B g sc In Eq. (6), R, identifies the sink/suce tem f the ate cmpnent at standad cnditins, heeas in Eqs. (8) and (9), B identifies the fmatin vlume fact f each phase, g cespnds t the gavitatinal acceleatin, and g c is the gavitatinal cnvesin fact. Finally, in Eq. (6), P c identifies the capillay pessue elatinship, descibed as P c( S ) = p p, (10) hee p cespnds t leic and p t aqueus phase pessues, espectively. F cases hee capillay pessue effects ae nn-negligible e emply an empiical elatinship intduced by Sanchez-Bujans (1996), namely, P c c e D p p = P ( 1 S )(1 S ), (11) hee Pc is a base capillay pessue f the pemable medium and e p is a cuvatue expnent detemined empiically fm lab expeiments. We adpt a deteministic pe la t gven the dependency f elative pemeability n ate satuatin. T that end, e define a dimensinless satuatin, S D D S Si =, (12) S S 1 i hee S i and S ae ieducible ate and esidual il satuatins, espectively. Relative pemeability functins ae then given by and e ( S ) k S, k = (13) D D e c c k e ( S ) k ( 1 S ), D = (14) hee k and ate-il elative pemeabilities, and D k ae the end-pint values f the e and ae ate and il satuatin expnents, espectively (Lake, 1989). SIMULATION OF ARRAY INDUCTION DATA Thee-dimensinal mdeling f inductin tls equies slving Maxell s equatins in a discete setting eithe by finite elements by finite diffeences. We make use f a 3D finite-diffeence fmulatin adapted f the mdeling f multiaay inductin tls. The simulatin cde uses the SLDMINV algithm (Duskin et al., 1999). This algithm is based n the Spectal Lanczs Decmpsitin Methd (SLDM) (Duskin and Knizhneman, 1994), using Kylv subspaces geneated fm the invese pes f the Maxell peat. The fequency dmain pblem f Maxell s equatins is cnsideed in the SLDMINV as flls: and e E + iω µ H = 0, (15) H ( σ + i ω ε) E = J, (16) heein negligible displacement cuents ae assumed, i.e., the dielectic cnstant, ε = 0. In the abve equatins, E, H, and J identify electic field, magnetic field, and electic suce density, espectively. The symbls σ = σ ( x, y, z) and µ dente hmic cnductivity and magnetic pemeability, espectively. Meve, ω is the angula fequency, and i is equal t 1. Substitutin f Eq. (15) leads t an equatin slely expessed in tems f electic field E, i.e., σ 1 1 µ E + iω E = i ω σ 1 J. (17) The spatial matix peat, A, f this last equatin is defined as 1 1 µ A = σ, (18) heeas a suce functin ϕ is defined as ϕ = i ωσ 1 J. (19) Equatin (17) then becmes ( A + iω I) E = ϕ, (20) 3

4 hee I dentes the identity matix. Bunday cnditins f the afedescibed electmagnetic pblem ae given by E n = 0, (21) Ω hee n is the nmal vect. The dmain f inteest is given by Ω = { x, y, z) : x x x, y min y y ( min max max, z min z zmax }. The slutin f Eq. (20) is implemented by appximating Eq. (17) ith a finite-diffeence 3 stencil in a 3D Catesian cdinate fame, R, and ith a staggeed gid (Yee, 1966). A hmgenizatin technique (Msk et al., 1999) is als used ithin the SLDMINV algithm that des nt equie the finite-diffeence gid t cnfm t the intefaces f the inhmgeneities. This hmgenizatin technique pvides a finitediffeence appximatin f cmplex gemeties ith the same degee f numeical accuacy as that f a finite element methd, but at a much le cmpute cst. In dipping fmatins, bed bundaies ae mdeled using a staicase defmatin. At l fequencies, the slutin f Eq. (20) can be eitten as E = iω ( A i ω I) A ϕ. (22) This expessin is nt nly highly efficient t simulate l fequency electmagnetic phenmena, but it als cnsideably educes the influence f spuius (nn-tatinal) numeical mdes. SLDMINV pvides additinal advantages ve standad finite-diffeence pcedues such as the ability t pduce esults f multiple fequencies in a single simulatin un, and the usage f matix peatins in eal aithmetic. The efficiency f the SLDMINV algithm elies n fast evaluatin f the invese pes f the stiffness peat, A. In tun, the latte peat is cmputed fm a decmpsitin f its cul-fee and divegence-fee pjectins. Slutins f these pjectins ae cmputed by making use f discete Fuie tansfms (DFT) and pecnditined cnjugate gadient (PCG) iteatins. Duskin et al. (1999) sh that the cnvegence ate f the SLDMINV methd inceases as fequency deceases, hich makes the SLDMINV paticulaly attactive f l-fequency applicatins. Extensive details f the SLDMINV ae pesented in Duskin et al. (1999). THE LINK BETWEEN SATURATION AND ELECTRICAL RESISTIVITY F simplicity, Achie s la (Achie, 1942) is used t establish a quantitative elatinship beteen fluid satuatin and electical esistivity. F the hydcabn-beaing vigin fmatin, Achie s elatinship is given by R ( ) Rt ( ) =. (23) m n φ S ( ) On the the hand, f the invaded zne ne has R Rmf () () = hee i = 1,..., n m S x. (24) φ S () x, i n x, i In the abve equatins, R, R, R, R, S and S x dente esistivities f vigin zne, fmatin ate (cnnate ate), mud-filtate, and invaded zne; and satuatins f vigin and invaded znes, espectively. The psity and satuatin expnents, m and n, espectively, ae empiical cnstants. The latte fm f Achie s la, given in Eq. (24), is used t descibe the tansfmatin fm ate satuatin t electical esistivity f multiple invasin znes. SENSITIVITY ANALYSIS We assess the sensitivity f AIT* measuements t the pesence f ate-base mud-filtate invasin in a hizntal ellbe f vaius cmbinatins f petphysical paametes and fluid ppeties. A schematic desciptin f the mdeling gemety adpted in u study is shn in Fig. 2. Opeating Assumptins f the Simulatin f Mud-filtate Invasin We assume that the adial Dacy speed at hich the mud-filtate fls int the fmatin, u f, is entiely cntlled by the mud ppeties. Hence, the vlume f filtate pe unit time enteing the fmatin can be calculated fm q mf t = 2π u h, (25) f hee is the ellbe adius, and h is the fmatin thickness in hich mud-filtate invasin takes place. Equatin (25) is a diect cnsequence f the lage hyddynamic esistance encunteed by filtate as it fls thugh the mud cake, in cmpaisn t the fmatin. The abve assumptin als flls fm the pesence f a mf x 4

5 mud cake, implying that the hizntal pemeability, k h, f the bed is abve sme minimum value. A minimum value f k h equied f the develpment f the mud cake depends pimaily n the diffeence beteen ellbe and fmatin pessues as ell as n the speed at hich the mud is pumped thugh the ellbe (me accuately, the shea stess at the ellbe all). Unde nminal cnditins, this equies k h t be geate than 10 t 20md (Dussan V. et al., 1994). Once the hizntal pemeability exceeds this value (a mud cake is pesent), then the specific value f u f is independent f h k. In this pape, e cnside the case in hich the ate f filtate invasin is cntlled by mudcake ppeties athe than by fmatin pemeability. Mud-filtate invasin ates ae calculated assuming a timeinvaiant invasin speed. A spatially unifm lcal invasin speed cnditin is impsed alng the behle all. We assume that the value f the 5 speed f adial filtate invasin is 5 10 cm / s, hich emains cnsistent ith the invasin speeds epted by Dussan V. et al. (1994). Synthetic Pemeable Medium Ou synthetic esevi is cnstucted using a 60ftlng hizntal sectin dilled thugh the cente f a 98ft thick and 98ft ide hydcabn-beaing zne satuated at ieducible ate satuatin. F multiphase fl simulatins, a cnstant ate inne bunday cnditin and a n-fl ute bunday cnditin ae impsed n the pemeable medium. Equatin (25) is used t calculate an invasin ate f 2.29bbl/d assuming an aveage ellbe adius f 0.3ft. We make use f an aveage equivalent ellbe adius given that the hizntal ellbe penetating thugh the pemeable medium is cnstucted ith a Catesian gid. A ( x, y, and z, espectively) Catesian gid is implemented t simulate detailed images f the nea-ellbe satuatin field. This gid cnsists f small spatial steps aund the ellbe and case steps aay fm the behle. Figue 3 shs a t-dimensinal (2D) csssectin f the finite-diffeence gid enfced f fluid fl simulatins. The gid is supeimpsed n an example f simulated satuatin field. Aveage initial fmatin pessue and tempeatue ae assumed t be 3000psi and 220F, espectively. Futheme, the assumptin is made that f the simulated pessue ange, the ck cmpessibility is small ith espect t the fluid cmpessibility. Hence, ttal cmpessibility is assumed t be slely a functin f the cmpessibilities f the satuating fluids. A pemeability-psity elatinship is als enfced in u fluid fl simulatins thugh an empiical celatin cmmnly epted in the ell-lgging liteatue, namely, φ 0.1*lg10 ( k ) , (26) = h hee hizntal pemeability is given in md. The latte empiical elatinship is typical f shaly sandstnes (Dussan V. et al., 1994). Paametes f Achie s la ae assumed as flls: cnnate ate esistivity, R = 0.56hm-m, mud-filtate esistivity, Rmf = 0.2hm-m, m = 2, and n = 2. We futhe assume a negligible salinity gadient ith espect t the ell lcatin. A 2D schematic f the hizntal css-sectin f the zne f inteest is shn in Fig. 4. The petphysical and fluid fl ppeties displayed in this figue summaize the anges f vaiability cnsideed in u sensitivity studies. Sensitivity Studies We analyze the sensitivity f mud-filtate invasin t a pemeability vaiatin in a hmgeneus and istpic pus fmatin (Study 1). Psity is cupled t hizntal pemeability by Eq. (26). Ppeties f bth mud-filtate and 30 API-il cnsideed in this study ae shn in Tables 1 and 2, espectively. We assume a ate-et ck fmatin and chaacteize the fluid displacement pcess as imbibitin gvened by a Cey-type mdel f elative pemeabilities cnsistent ith the ck ettability {see Eqs. (13) and (14)}. In additin, e neglect hysteesis in elative pemeability. Capillay effects ae als assumed t be negligible. The tp-mst panels f Fig. 5 ae css-sectins f the simulated nea ellbe satuatin fields f ck fmatin pemeabilities f 100md, 500md, and 1000md, espectively. A cl gadient fm dak blue t ed is used t descibe the ate satuatin ange S ) fm ( ieducible ate satuatin ( S i ) t ate satuatin at esidual il satuatin { 1 S )}. The ( ellbe egin is depicted ith a cicle in the css-sectins. We utilize multiple degees f smthing n the pst-simulatin satuatin field. Cnsequently, multiple levels f gemetical cmplexity ae cnsideed in the esistivity dmain subsequent t the applicatin f Achie s la n 5

6 the spatially smthed satuatin fields. The mid and bttm panels f Fig. 5 display the simplest and the mst detailed smthed satuatin images, espectively. Figues 12(a) thugh 12(c) sh the simulated electmagnetic espnses cespnding t ck fmatin pemeabilities f 100md, 500md, and 1000md, espectively. Electmagnetic espnses f each f the five AIT* channels ae pesented f the simplest and mst detailed invaded egin gemety cnsideed as a esult f the spatial smthing pcedue. F cmpaisn, the esistivity f the uninvaded zne, R t, is als shn in the AIT* sensitivity plts. All fluid-fl sensitivity simulatins descibed in this pape ae pesented as time snapshts taken n the 3 d day f mud-filtate invasin and subject t the pescibed invasin ate. A secnd sensitivity study is intended t assess the effect f a vaiatin in il type satuating the zne f inteest. T this end, e simulated mud-filtate invasin int heavy, intemediate, and light hydcabn-beaing fmatins. The liquid hydcabn phase is chaacteized using typical API gavities f heavy, intemediate, and light ils. Vaiatin f API inheently embdies vaiatins in density, fmatin vlume fact, cmpessibility, and viscsity f the hydcabn phase. Fm the displacement viepint, this study incpates t gvening paametes, namely, density cntast and viscsity ati, hich in tun cntl the effect f gavity and phase mbilities. Tables 1 and 2 ae ganized summaies f the fluid paametes cnsideed in u sensitivity study. T values f hmgeneus and istpic pemeability, k, f the ck ae als cnsideed in this study: a high k f 500md (Study 2), and a mdeate k f 100md (Study 3). Remaining gemetical and petphysical ppeties ae assumed t be equal t thse descibed in Study 1 (base case). Figues 6 and 7 sh the simulated satuatin fields and thei spatially smthed cuntepats f Studies 2 and 3, espectively. AIT* sensitivities ae shn in Figs. 13(a) thugh 13(c) f Study 2 and 14(a) thugh 14(c) f Study 3, espectively. The impact f pemeability anistpy n mudfiltate invasin, and cnsequently, n the AIT* espnse is cnsideed in Study 4. Pemeability anistpy ati, R anis, is defined as the ati f hizntal t vetical pemeability, k h k v. In Study 4, e analyzed cases hee R is equal t anis 1, 10, and 100, espectively. We futhe assumed a typical hizntal pemeability, k h, f 100md, and assigned the vetical pemeability, k v, values f 100md, 10md, and 1md, espectively. The emaining fluid and petphysical paametes ae equal t thse f the base case. Figue 8 displays the ensuing fluid satuatin plts, and Figs. 15(a) thugh 15(c) descibe the sensitivity f the AIT* espnse t pemeability anistpy. Sensitivity f the AIT* espnse t a vaiatin in ck ettability is examined by mdifying the mdel-based Cey type elative pemeability, k, cuves used f fluid fl simulatins {see Eqs. (13) and (14)}. Typical end-pint elative pemeabilities, satuatins, paametes, i k and k, and esidual S and S, alng ith cuvatue e and e, ae used f a ate-, intemediate- and il-et ck. Additinally, a typical imbitin capillay pessue, P c, elatinship is emplyed t assess the effect f capillay pessue n filtate invasin f the case f a ateet ck. We assume t hmgeneus and istpic pemeabilities, k : a high k f 500md (Study 5), and a mdeate k f 100md (Study 6). Figues 9 and 10 sh the esults f fluid fl simulatins in the fm f 2D css-sectin f nea-ellbe fluid satuatin at the 3 d day f invasin f Studies 5 and 6, espectively. The ange f fluid satuatin is unique t each case f ck ettability ing t the vaiability in endpint satuatins. Hence, the cl gadient used in the satuatin plts shuld be intepeted independently f each case. Figue 11 shs the values f k as ell as the P c elatinship used in the fluid fl simulatins. The cespnding AIT* espnses ae shn in Figs. 16(a) thugh 16(d) f Study 5 and Figs. 17(a) thugh 17(d) f Study 6, espectively. Intepetatin f the Simulatin Results Seveal significant bsevatins can be made fm the intepetatin f the simulatin f bth mudfiltate invasin (satuatin distibutin and gemety) and penhle electmagnetic lgging (AIT*) espnse. The simulated nea-ellbe satuatin fields clealy indicate that inceasing values f pemeability density cntast acceleate fluid mvement and incease the depth f sagging in the vetical diectin at the bttm f the hizntal ell. Heve, the extent f invasin becmes nae at the sides and, especially, at the tp f the hizntal ellbe. 6

7 Highe values f mud-filtate satuatin, chaacteized by less spatial spead, ae bseved almst symmetically aund the ellbe. This leads t cylindical invasin shapes f the cases in hich sagging at the bttm f the ell is less significant. Invasin gemeties evlve fm simple cylindical t ff-centeed cylindical shapes. Theeafte, thse shapes evlve int elliptical and skeed elliptical fms and even spatially me cmplex pisms as the value f pemeability density cntast inceases. Time evlutin f invasin fnts diffes substantially fm the hmgeneus and istpic case as the anistpy ati becmes me sevee. Invasin fnts evlve fm cylindical t ellbe-centeed elliptical pisms that extend tad the sides f the ellbe f inceasing values f anistpy ati. Spatial distibutins f mud-filtate assciated ith anistpic ck fmatins ae chaacteized by symmetical, ideextent fnts n the sides, and spatially cncentated fnts at the tp and bttm f the ellbe. Typically, the highe the anistpy ati, the me skeed the shape f the elliptical invasin pism. This spatial distibutin f mudfiltate is analgus t that esulting fm the fl f filtate alng a hizntal factue intesecting the hizntal ellbe. Inceasing the anistpy ati suppesses the effect f density cntast high pemeability by educing the fluid mvement in the vetical diectin at the bttm f the behle. Vaiability in the ck s ettability diectly affects displacement efficiency theeby causing significant vaiatins in aveage satuatins and invasin gemeties aund the ellbe. The distibutins f fluid satuatin shn in Studies 5 and 6 indicate lage invasin znes ith inceasingly highe il satuatins as the ck ettability is vaied fm ate- t il-et. This esult is cnsistent ith the physics f immiscible displacement dictated by the elatinships f elative pemeability. The backgund effect f the vaiability in abslute pemeability is eslved by cmpaing the fluid satuatin plts f Studies 5 and 6. We bseve that the veall invasin gemety is detemined by abslute pemeability as it cntls the sagging at the bttm f the ell. Heve, the spatial distibutin f fluid satuatin ithin the invaded zne emains a stng functin f ettability. We examine a limited numbe f cases intended t assess capillay pessue effects. It is bseved that capillay pessue suppesses the effects f bth density cntast and high pemeability. It als dispeses the satuatin fnt, and educes the dis placement efficiency, theeby causing lage ellbe-centeed cylindical invasins ith highe il satuatins ithin the invaded zne. Signatues f bth fluid and petphysical paametes can be bseved n the AIT* tl espnse. A chaacteistic incease in the shalle esistivity channel is cnsistently bseved as a functin f inceasing filtate sagging at the bttm f the behle. The espnse fm the shalle esistivity channels becmes clse t the esistivity f the vigin zne,, t R as filtate sagging inceases at the bttm f the ell. Quite bviusly, the AIT* espnse is sensitive t the aveage medium ppeties aund the behle. Uninvaded znes at the tp, and slightly invaded znes at the sides f the ellbe cnstuctively influence the tl espnse. This causes a eductin in the deviatin f the esistivity espnse due t sevee invasin at the bttm f the ell, heeupn the AIT* espnse yields esistivity values clse t thse f the vigin zne. Cnsequently, petphysical and fluid fl cnditins in the pesence f high pemeability and/ high gavity cntasts beteen invading and in-situ fluids (f a elatively l pemeability anistpy ati) have a distinct impint n the shalle AIT* esistivities. Inceasing the anistpy ati suppesses the invasin at the tp and at the bttm f the ell and esults in deepe hizntal invasin. As a functin f inceasing anistpy ati, the shalle AIT* channels mntnically yield esistivity values clse t thse f the vigin zne. An incease in the anistpy ati causes a thinning in the lense-shaped distibutin f hizntal mud-filtate invasin. Uninvaded zne satuatins at the tp and at the bttm f the ell stngly influence the AIT* tl espnse as a functin f inceasing anistpy. Aveage hydcabn satuatin aund the ellbe inceases and the veall invasin zne gs lage as the ck s ettability is shifted fm ate- t intemediate- and finally t il-et. The cespnding AIT* espnse cnsistently yields highe esistivity values fm shalle channels as a functin f the abve mentined 7

8 ettability shift. An enlagement f the shape f the invasin zne is bseved as me deep-sensing channels ae influenced by the invasin pcess. The fnt-dispesing effect pduced by capillay pessue yields high aveage il satuatins as ell as ellbe-centeed deep invasins. Again, shall AIT* channels yield esistivity values clse t, t R but the effect f deep invasin emains bsevable in the deep-sensing channels. Despite the advese influence f capillay pessue, f all f the investigated cases, the deepest penetating AIT* channel cntinues t exhibit a stng influence f the vigin zne esistivity, R t. Pi knledge f ne me f the cntlling petphysical and fluid paametes tgethe ith the AIT* lg espnse can significantly enhance the intepetatin f bth the extent and the gemety f the invasin zne. In additin, f a significant numbe f investigated cases, the AIT* espnse emains sensitive t the gss gemety f invasin athe than t the detailed gemetical featues f the invaded zne. Despite this insensitivity t the detailed gemety f invasin, the AIT* still exhibits sufficient sensitivity t enable a cectin f invasin effects n the deepe channels. CONCLUSIONS The flling cnclusins can be dan fm the sensitivity study pesented in this pape: Petphysical and fluid paametes play a cental le in detemining (a) the time evlutin f the shape and lateal extent f the mud-filtate invasin and (b) the distibutin f satuatins ithin the invaded zne f a hizntal ell. Cnsequently, undestanding the influence f petphysical and fluid paametes in aay inductin measuements acquied in a hizntal ell equies f a igus quantitative intepetatin appach that addesses multiphase fl phenmena. It has been fund that the le played by petphysical and fluid paametes in the time evlutin f mud-filtate invasin can be me significant in the case f hizntal ells than in the case f vetical ells. Aay inductin tls exhibit limited sensitivity t seveal f the cmplex gemetical featues f the satuatin distibutin aund a hizntal behle. Theefe, it has been fund that ften simple paametic epesentatins f bth invasin gemety and invaded zne satuatins can be utilized t quantitatively intepet aay inductin measuements int fluid satuatins in the vigin zne. Test cases designed t quantify the effect f capillay pessue n mud-filtate invasin sh that deep invasin can ccu in the pesence f elatively l mud-filtate satuatins. Clealy, quantifying the capillay pessue effects n the AIT* espnse equies f a me extensive study than the ne pesented in this pape. F all f the test cases cnsideed in u sensitivity study, including thse aimed at quantifying the effect f capillay pessue, the deepest penetating AIT* channel emains stngly influenced by the vigin zne s esistivity, R t. Hence, e cnclude that the AIT* 5 channels embdy sufficient infmatin t enable accuate and bust estimatin f R thugh nnlinea invesin. ACKNOWLEDGEMENTS We uld like t expess u gatitude t Schlumbege f the funding f this k thugh a summe intenship psitin pvided t Fauk. O. Alpak. The auths als ish t thank Ds. Vladimi Duskin, Babaa Andesn, and Ping Lee, f valuable discussins and suppt thughut the cmpletin f this pject. REFERENCES 1. Andesn, B.I., Babe, T., Duskin, V., Lee, P., Dussan V., E.B., Knizhneman, L., and Davydycheva, S.: The espnse f multiaay inductin tls in highly dipping fmatins ith invasin and in abitay 3D gemeties, The Lg Analyst, 40 (5), p , Septembe-Octbe Achie, G.E.: The electical esistivity lg as an aid in detemining sme esevi chaacteistics, Petleum Tansactins f the AIME, 146, p , Aziz, K. and Settai, A.: Petleum Resevi Simulatin, Applied Science Publishes Ltd., Lndn (1979). 4. Duskin, V. and Knizhneman, L.: Spectal appach t slving thee-dimensinal Maxell s diffusin equatins in the time and fequency dmains, Radi Science, 29 (4), p , Duskin, V., Knizhneman, L., and Lee, P.: A ne spectal Lanczs decmpsitin methd f inductin mdeling in abitay 3D gemety, Gephysics, 64 (3), 1999, Dussan V., E.B., Andesn, B.I., and Auzeais, F.: Estimating vetical pemeability fm esistivity lgs, pape pesented at the 35 th SPWLA Lgging Sympsium, Tulsa, Oklahma, June 19-22, Msk, S., Duskin, V., Habashy, T., Lee, P., and Davydycheva, S.: A finite diffeence scheme f elliptic equatins ith ugh cefficients using a Catesian gid nncnfming t intefaces, SIAM Junal n Numeical Analysis, 36 (2), p , Lake, L.W.: Enhanced Oil Recvey, Pentice Hall, Engled Cliffs, p , Ramakishnan, T.S. and Wilkinsn, D.J.: Fmatin pducibility and factinal fl cuves fm adial esistivity vaiatin caused by dilling fluid invasin, Phys. Fluids, 9 (4), p , Apil Ramakishnan, T.S. and Wilkinsn, D.J.: Wate-cut and factinal fl lgs fm aay-inductin measuements, SPE Resevi Eval. & Eng. 2 (1), p , Febuay t 8

9 11. Sanchez-Bujans, J. L.: Scaling paametes f chaacteizing atefld ecvey fm anistpic matix blcks in natually factued esevis, PhD dissetatin, U. f Texas, Austin, Yee, K.S.: Numeical slutin f initial bunday value pblems invlving Maxell s equatins in istpic media, IEEE Tans. Antennas Pp. AP-14, , Zhang, J-H., Hu, Q., and Liu, Z-H.: Estimatin f tue fmatin esistivity and ate satuatin ith a time-lapse inductin lgging methd, The Lg Analyst, 40 (2), p , ABOUT THE AUTHORS Fauk O. Alpak is a Reseach Assistant and a Ph.D. candidate in the Depatment f Petleum Engineeing at The Univesity f Texas at Austin. He eceived a Maste f Science degee fm The Univesity f Texas at Austin in 1999 in Petleum Engineeing. Elisabeth Dussan V. is a Scientific Advis in the Real-Time Mdeling and Invesin Theme at Schlumbege-Dll Reseach in Ridgefield, CT. He degees ae in the field f fluid mechanics. Since jining Schlumbege in 1985 she has ked n diffeent methds f detemining pemeability anistpy and n the dynamics f filtate invasin. Desciptin f the Pemeable Medium and Fluids via Paametes that Cntl Multiphase Fluid Fl (MUD-FILTRATE INVASION) FLUID FLOW SIMULATION ECLIPSE 100 Satuatin Field at t = t lg Linking S int R ARCHIE S LAW Rmf R x,i = m n φ S x, i Desciptin f the Pemeable Medium and Fluids via Paametes that Cntl Aay Inductin Respnse AIT RESPONSE SIMULATION SLDMINV AIT Resistivity Respnse Vaiables: Petphysical and fluid ppeties that gven multiphase fl physics Sensitivity f AIT Sensitivity t t multiphase fl satuatin field physics desciptin Figue 1. Oganizatin fl chat used t quantitatively assess the sensitivity f aay inductin tls t mud-filtate invasin in hizntal ells. Taek M. Habashy eceived a Ph.D. fm MIT in 1983 in Electical Engineeing. Duing he as a Reseach Assciate in the Depatment f Electical Engineeing at MIT. Since Septembe 1983, he has been ith Schlumbege-Dll Reseach, and is n the Theme Manage f the Real-Time Mdeling and Invesin Theme and a Scientific Advis cnducting eseach in electmagnetic lgging tls, invese scatteing they, and numeical analysis. He seved as an Assciate Edit f Radi Science and is cuently a membe f the Editial Bads f Invese Pblems and the Junal f Electmagnetic Waves and Applicatins. He is a Fell f the Institute f Physics and IEEE. Cals Tes-Vedín eceived a Ph.D. in Engineeing Gescience fm the Univesity f Califnia, Bekeley, in Duing he held the psitin f Reseach Scientist ith Schlumbege-Dll Reseach. Fm , he as Resevi Specialist and Technlgy Champin ith YPF (Buens Aies, Agentina). Since 1999, he is an Assistant Pfess ith the Depatment f Petleum and Gesystems Engineeing f The Univesity f Texas at Austin, hee he cnducts eseach in fmatin evaluatin and integated esevi chaacteizatin. He has seved as Guest Edit f Radi Science, and is cuently a membe f the Editial Bad f the Junal f Electmagnetic Waves and Applicatins, and an assciate edit f Petphysics (SPWLA). Figue 2. Gemetical mdel cnsideed f the simulatin f t-phase fl (simultaneus fl f leic and aqueus phases) and electmagnetic inductin (adpted fm Andesn et al., 1999). Table 1. Summay f mud-filtate (aqueus phase) ppeties at a pessue f 3000psi and a tempeatue f 220F. Fmatin Vlume Fact, B, (bbl/stb) Density, ρ, (lbm/cuft) Cmpessibility, C, (1E-06 x psi -1 ) Viscsity, µ, (cp) Table 2. Summay f hydcabn (leic phase) ppeties at a pessue f 3000psi and a tempeatue f 220F. API ρ B c = c fl µ ρ = ρ ρ ρ µ (lbm/cuft) (bbl/stb) (x 10-6 psi -1 ) (cp) (lbm/cuft) (g/cm 3 ) µ

10 Figue 3. T-dimensinal css-sectin pependicula t the behle axis f the finite-diffeence gid cnstucted f the simulatin f t-phase fl. The simulatin gid is shn supeimpsed t an example ate satuatin field. Figue 6. Study 2: Sensitivity f the nea-ellbe fluid satuatin field t a vaiatin in il type ithin a highpemeability fmatin. The tp panels ae detailed images f nea-ellbe ate satuatin simulated f vaius API ils. Oil types ae chaacteized by the values f fluid density, viscsity, and cmpessibility shn in Table 2. Refe t Fig. 5 f futhe details n the pltting cnventins. Figue 4. T-dimensinal css-sectin f the zne f inteest tavesed by a hizntal ell. The petphysical and fluid fl ppeties displayed in the figue summaize the anges f vaiability cnsideed in u sensitivity studies. Figue 7. Study 3: Sensitivity f the nea-ellbe fluid satuatin field t a vaiatin in il type ithin a mdeatepemeability fmatin. Cmpae ith Fig. 6 (esults f Study 2). The tp panels ae detailed images f nea-ellbe ate satuatin simulated f vaius API ils. Refe t Fig. 5 f futhe details n the pltting cnventins. Figue 5. Study 1: Sensitivity f the nea-ellbe fluid satuatin field t a vaiatin in abslute pemeability. Psity is empiically cupled t abslute pemeability. The tp panels ae detailed images f nea-ellbe ate satuatin simulated f vaius values f abslute pemeability. Mid and bttm panels display the simplest and the mst detailed smthed images, espectively, used t tansfm nea-ellbe satuatins int electical esistivities. Figue 8. Study 4: Sensitivity f the nea-ellbe fluid satuatin field t a vaiatin in pemeability anistpy. The tp panels ae detailed images f nea-ellbe ate satuatin simulated f vaius pemeability anistpy atis. Hee, anistpy ati is defined as the ati f hizntal t vetical abslute pemeabilities. Refe t Fig. 5 f futhe details n the pltting cnventins. 10

11 Figue 9. Study 5: Sensitivity f the nea-ellbe fluid satuatin field t a vaiatin in the ettability chaacteistics f a high -pemeability fmatin. The tp panels ae detailed images f nea-ellbe ate satuatin simulated f typical etting chaacteistics f the esevi ck. Refe t Fig. 5 f futhe details n the pltting cnventins. Figue 10. Study 6: Sensitivity f the nea-ellbe fluid satuatin field t a vaiatin in the ettability chaacteistics f a mdeate-pemeability fmatin. Cmpae ith Fig. 9 (esults f Study 5). The tp panels ae detailed images f nea-ellbe ate satuatin simulated f typical etting chaacteistics f the esevi ck. Refe t Fig. 5 f futhe details n the pltting cnventins. ate-et k intemediate-et k il-et k ate-et P c Figue 11. Wettability chaacteistics f the pemeable medium. The figues sh h the ettability ppeties ae changed by mdifying the elative pemeability elatinships used f fluid fl simulatins. Typical end-pint elative pemeabilities and esidual satuatins alng ith cuvatue paametes ae used t descibe ate-, intemediate- and il-et cks. An additinal plt descibes the imbitin capillay pessue elatinship emplyed f fluid fl simulatin in the ate-et case. 11

12 (a) (a) (b) (b) (c) Figue 12. Simulated electmagnetic espnses f Study 1- Cases 1, 2, and 3. Five AIT* electical esistivity channels ae shn f each f the panels (a), (b), and (c). (c) Figue 13. Simulated electmagnetic espnses f Study 2- Cases 1, 2, and 3. Five AIT* electical esistivity channels ae shn f each f the panels (a), (b), and (c). 12

13 (a) (a) (b) (b) (c) Figue 14. Simulated electmagnetic espnses f Study 3- Cases 1, 2, and 3. Five AIT* electical esistivity channels ae shn f each f the panels (a), (b), and (c). (c) Figue 15. Simulated electmagnetic espnses f Study 4- Cases 1, 2, and 3. Five AIT* electical esistivity channels ae shn f each f the panels (a), (b), and (c). 13

14 (a) (a) (b) (b) (c) (c) (d) Figue 16. Simulated electmagnetic espnses f Study 5- Cases 1, 2, 3, and 4. Five AIT* electical esistivity channels ae shn f each f the panels (a), (b), (c), and (d). (d) Figue 17. Simulated electmagnetic espnses f Study 6- Cases 1, 2, 3, and 4. Five AIT* electical esistivity channels ae shn f each f the panels (a), (b), (c), and (d). 14