The Influence of the Model Fractal Parameters on the Electromagnetic Response in Environment with Cylindrical Layers

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1 Intenational Jounal of Geosciences, 07, 8, ISSN Online: ISSN Pint: The Influence of the Model Factal Paametes on the Electomagnetic Response in Envionment with Cylindical Layes Valci João da Cunha Faias, Bígida Ramati Peeia da Rocha, Alice Nabiça Moes Faculty of Statistics, Fedeal Univesity of Paá, Belém, Bail Pos-Gaduate Pogam in Electical Engineeing, Fedeal Univesity of Paá, Belém, Bail How to cite this pape: Faias, V.J.C., da Rocha, B.R.P. and Moes, A.N. (07) The Influence of the Model Factal Paametes on the Electomagnetic Response in Envionment with Cylindical Layes. Intenational Jounal of Geosciences, 8, Received: Decembe 6, 06 Accepted: Mach 8, 07 Published: Mach 3, 07 Copyight 07 by authos and Scientific Reseach Publishing Inc. This wok is licensed unde the Ceative Commons Attibution Intenational License (CC BY 4.0). Open Access Abstact The induced polaiation esponse of an envionment pesenting cylindical layes was obtained. The factal model fo complex esistivity was employed as an intinsic popety of the polaiable layes. The influence of the model factal paametes on the electomagnetic esponse was investigated. The esults demonstated that the factal paametes dominate the appaent esistivity phase esponse; measuements of the induced polaiation data allow fo the detemination of the factal popeties of the envionment without noticeable electomagnetic coupling effects at fequencies below 0 4 H. Keywods Factal Model, Induced Polaiation, Complex Resistivity, Well Log. Intoduction The induced polaiation effect has an electochemical oigin and is usually associated with geological and biological envionments [] [] [3]. As a consequence of this effect, electical esistivity values in these envionments ae complex and fequency-dependent. In geophysics, the induced polaiation method uses fact that the constituent paametes of ocks (conductivity and pemittivity) ae fequency-dependent to cay out pospecting activities. This method was oiginally applied to the pospecting of disseminated oes, and has gadually evolved ove the yeas, cuently being used in mineal discimination [4] and envionmental studies [5] and [6]. The quantitative intepetation of field induced polaiation data is a difficult task due to the factal natue of geological envionments and the inductive DOI: 0.436/ijg Mach 3, 07

2 coupling caused by electomagnetic inteactions between the envionment and the electode aays used fo cuent injection and potential measuements. The intepetation of these kinds of data equies a physical model to explain the behavio of a polaiable envionment in an ample fequency ange. Seveal elaxation models have been poposed to descibe the electical polaiation of ocks, in the woks of Debye [7], Cole-Cole [8], Davidson and Cole [9] and Dias [0], each taking into account a cetain specific chaacteistic fo a given fequency ange, limited to 0 H. Relaxation models demonstate the geneal behavio of the amplitude spectum and the complex esistivity phase (conductivity) at diffeent fequencies fo diffeent types of mateials. The most widely used model is the Cole-Cole model, which does not, howeve, conside the factal natue of the envionment. Rocha [] developed a model that consides the factal effects of poous sufaces and includes ock volume esponse, namely the factal model fo complex esistivity. This model accounts fo the electical popeties of ocks at a highe fequency ange than taditional models. The intoduction of the oughness facto in this model allows fo the investigation of ock textue, which is vey impotant when attempting to descibe the electical behavio of ocks. This means that paametes epesenting the factal geomety of the envionment exist, which may be, in tun, elated to ock textue. With this, it is possible to obtain impotant and accuate geological infomation of the subsuface fom electical data obtained on the teain suface. Rocha [] and Rocha & Habshy [3] detemined the esponse of a teain pesenting thee hoiontal layes, with the second laye polaiable with its intinsic popeties given by the factal model fo complex esistivity of [] and analyed the induced polaiation esponse. These authos obseved that the paametes elated to the factal geomety of the model dominate the phase esponse of the appaent complex esistivity at low fequencies, and also found that the factal exponent does not depend on the electical popeties of the fluid filling the ock cavities. The fact that the factal exponent is independent fom the electical esistivity of the pecolating solutions avoids any influence of the invasion one in the electic pofiling of wells. Thus, it is inteesting to investigate the esponse of a polaiable medium in well envionments by applying the factal model fo complex esistivity. Faias et al. [] [3] simulated the factal model fo complex esistivity as being an intinsic electical popety of hoiontal envionments with supeficial and volumetic fomations (-D and 3-D geological models), with applications fo both contaminated and non-contaminated envionments. The esults demonstated that anomalies ae well-detected and obsevable by images of the paamete distibution of the factal model, being an altenative in the detection of anomalies in the geologic envionment, such as in the study of envionmental contamination. The factal complex esistivity model, howeve, has not yet been applied as an intinsic electical popety in the analysis of the polaiation e- 350

3 sponse of cylindical envionments. The main aim of electical well pofiling is to estimate the electical esistivity of the geological fomation whee the well is inseted. Howeve, the esponse of the esistivity pofiling is influenced by the esistivity of the fomation itself, as well as by the invaded one, which is geneated duing the dilling pocess. Theefoe, the effects of this invaded aea should be avoided. In the pesent study, the factal model fo complex esistivity [] is employed as an intinsic electical popety of an envionment with cylindical layes (the well, invaded one and fomation) to evaluate the influence of the paametes of the factal model in the induced polaiation esponse in this geological geomety. The model paametes epesent the factal geomety of the envionment which, as pesented peviously, can be elated to the textue of the ocks in the analyed envionment.. The Factal Model Repesenting the time dependence of the electic field as e iωt, the expession poposed by [4] fo the complex esistivity ρ( ω ) is: ρ( ω) = ρ o m γh + u + δ ( + v) whee ρ o is the DC esistivity of the mateial; m is the chageability defined by [4]; δ is the paamete that elates the esistivity of the conductive gains blocking the poes of the geologic envionment to the DC esistivity of the ock η matix; γ h = ( + iωτ o) ; u = iωτ ( + v) ; v = ( iωτ f ) ; τ is the elaxation time constant elated to the double-laye oscillations; τ o is the elaxation time constant associated with the mateial as a whole; τ f is the time of factal elaxation, elated to the time involved in the chage and enegy tansfe in the ough intefaces; and η is the paamete diectly elated to the factal geomety of the envionment, detemined by the type and distibution of the mineal that causes the polaiation at low fequencies. Some typical values of the factal model paametes fo complex esistivity 6 cited by [], ae: ρ o = 0 Ω m ; m = 0,5 ; δ = ; η = 0,5 ; τ = 0 s ; 3 = 0 s and = 0 s. τ f τ o 3. Induced Polaiation Response in a Statified Cylindical Envionment Nomally a fou-electode configuation is used to measue the complex esistivity of a geological envionment. An electic cuent is intoduced into the envionment via an electode pai (A and B) and the voltage is measued by the othe electode pai (M and N). Figue illustates the fou-electode configuation used in the pesent study to detemine the appaent esistivity on an envionment pesenting cylindical layes. () 35

4 Figue. Illustation of the fou-electode configuation applied to a statified cylindical envionment. To calculate the potential measued by the eceive electode pai, the electomagnetic poblem fo a fou-electode configuation must be solved. Fom Maxwell equations and assuming a time dependence of the i t e ω type, we have: E = iωµ H () H = iωε E + E + J S (3) whee J S is the cuent density due to the souce; ε is the effective dielectic constant and μ is the pemeability of the envionment and is appoximated by the pemeability of vacuum. Combining the conductivity ( ) and displacement facto ( iωε ), the cuent density can be denoted as: and (3) can be denoted as: J = E (4) H = J + J S (5) When applying the divegence opeato to () and (5), the following equations ae obtained: H = 0 (6) J = J S (7) whee the left side of (7) is a esult of the chage accumulation caused by the injected cuent. When obseving (6), the Maxwell equations can be displayed in tems of a vecto potential A and scala potential φ H = A (8) E = iωµ A φ (9) obseving (4), (5), (8) and (9), and consideing the condition o 35

5 A = φ (0) whee = iωε is the complex conductivity. Thus, the wave equations below ae obtained: with k A+ k A = J S () + k = J S () φ φ = iωµ. Specifying the cuent density in cylindical coodinates, with J S ( ) ( ) ( ) Iδ u u = e (3) whee u is a Heaviside function, I is the cuent intensity, and ae the positions of the cuent electodes (A e B) and e is the unit vecto in the diection. Consideing (8), (9) and (0) and the fact that the vecto potential is of the (0,0,A ) fom, the following equations ae obtained: With A E = iωµ oa + (4) A H θ = ρ (5) A being the solution fo () in the intenal egion of the well, then ( ) ( ) ( ) Iδ u u A + k A = (6) is valid fo a, whee a is the well adius. If A is the solution to () in the invaded one, then + = (7) A ka 0 is valid fo a b, whee b is the adius of the invaded one. In the same manne, egading the fomation, with a solution A 3, the following equation is obtained: + = (8) A3 k3a3 0 fo b. The tangential components of E and H ae continuous in the intefaces, o E = E and H = H in = a θ θ and E = E and H = H in = b 3 θ 3θ thus, by using (4) and (5) the bounday conditions to solve (6), (7) and (8) ae obtained. A A iwµ A iwµ A o + = o + = a = a A A = = a = a (9) (0) 353

6 A A iwµ A iwµ A 3 o + = o 3 + = b = b A A3 = = b = b () () In ode to solve Equations (6), (7) and (8) subject to the bounday Equa- G, tions (9), (0), () and (), it is convenient to intoduce the Geen ( ) equation, that satisfies the following equations: ( ) ( ) Iδ δ G + ug = fo a (3) G + ug = 0 fo a b (4) subject to the bounday conditions below: G3 + ug 3 3 = 0 fo b (5) G G iwµ G iwµ G o + = o + = a = a G G = = a = a G G iwµ G iwµ G 3 o + = o 3 + = b = b G G3 = = b = b The solutions to Equations (3), (4) and (5) ae given by: Since (6) (7) (8) (9) iur Ie G = + A( k ) ( ) ( ) ( ) cos ( ) d 4π 0 Ko k + B k Io k k k R (30) G is egula in 0 A k =, thus: =, then ( ) 0 iur Ie G = + B( k ) ( ) cos ( ) d 4π 0 Io k k k R (3) Using the identity (5) [5]: e R iur = K ( k ) cos k ( ) dk π 0 o in Equation (3), the following equation is obtained: I G = K k k k + B k I k k k π ( ) cos ( ) d ( ) ( ) cos ( ) d (3) 0 o 0 o valid fo a ; ( ) ( ) ( ) ( ) cos ( ) G = C k K k + D k I k k dk (33) 0 o o valid fo a b; and in b, whee I o and ( ) ( ) cos ( ) G = E k K k k dk (34) 3 0 o 3 K o ae modified Bessel functions of ode eo of the k j =,, 3 and R ae given by: fist and second species, espectively, and ( ) j 354

7 j j ( ) k = k u R =. The vecto potential function Aj (, ) Geen function ( G l ) : ( ) = ( ) is detemined by the integal of the Al, Gj, d (35) Imposing bounday conditions (6)-(9) in Equations (3), (33) and (34), functions B, C, D and E ae detemined. In this way, the following system of equation is obtained: Ik k Io( k a) B k Ko( ka) C Io( ka) D K o( k a) ; + = π I k I( k a) B+ k K( ka) C I( ka) D = k K( k a) ; π k Ko( kb) C + Io( kb) D k3ko( k3b) E = 0; 3 e k K( kb) C I( kb) D k3k( k3b) E = 0 to calculate the induced potential diffeence measued by electodes M and N (Figue ). Thus only the vecto potential A (, ) must be obtained, which is associated with the function B( k ). Solving the above system, the following equations ae obtained: ( ) B k I FH FH = π FH + FH ( + ) 3 4 k F = K k a K k a K k a K k a ( ) ( ) ( ) ( ) o o k k F = K k a I k a + K k a I k a ( ) ( ) ( ) ( ) o o k k F = I k a K k a + I k a K k a ( ) ( ) ( ) ( ) 3 o o k k F = I k a I k a I k a I k a ( ) ( ) ( ) ( ) 4 o o k k H = K k b I k b + K k b I k b 3 ( ) ( ) ( ) ( ) 3 o o 3 3 k k3 = o 3 o 3 3 k ( ) ( ) ( ) ( ) H K k b K k b K k b K k b When combining Equations (3) and (35), A (, ) can be witten as: I dk A(, ) = ( ) ( ) ( ) sen ( ) sen ( ) 0 Ko k + B k Io k k π k (37) With B( k ) given by Equation (36). The potential diffeence measued by electodes M and N is given by the line integal: V(, ) E(, ) dl = (36) 355

8 with dl = de, so A V(, ) = E(, ) d= + iwµ o A thus, k I V (, ) = K ( k ) B( k ) I ( k ) cos ( ) cos ( ) dk 0 o o k π (38) The potential V MN is calculated as follows: (, ) (, ) V = V V MN o N o M whee o is the adial position of the electodes in the well and and M N ae the vetical positions of the potential electodes. The integal of Equation (38) is solved by quadatue technique [6] o by digital filtes [7]. A simila esult was found by [8] in the study of the anisotopy effect on esistivity measuements in wells and by [9] in the study of a dynamic model fo esistivity and induced polaiation data in wells. 4. Results The induced polaiation esponses of the two geological two situations wee obtained by applying Equation (38): ) the envionment pesenting two cylindical layes (the well and the fomation); ) the envionment pesenting thee cylindical layes (the well, the invaded one and the fomation). The esistivity of the mud fo the two geometies was of Ω m when disegading the polaiation effect. The distances between the electodes, in metes, wee of 0.4, 6., 0.9 and 6.59 fo AM, NA, BN and BM, espectively. The default value fo the well adius was of 0 cm. In ode to analye the influence of paametes η, m, δ, τ and τ f of the factal model, the simulations wee caied out fo thee diffeent values fo each of these paametes, and when vaiations in a cetain paamete occued the othes assumed the typical values descibed above. 4.. Envionment Pesenting Two Layes Figue displays the induced polaiation esponse fo a well with only two cylindical layes (mud and fomation). The following values wee used: 0.5, 0.5 and 0.75 fo the factal exponent η; 0.5, 0.5 and 0.75 fo chageability; 0., and 0 fo paamete δ ; 0 9 s, 0 6 s and 0 3 s 3 fo the time constant τ and 0 4 s, 0 3 s and 0 fo the factal time constant τ. f It is obseved fom Figue that the factal model fo complex esistivity can be used in a wide fequency ange at envionment with cylindical layes. As in the case of hoiontal layes [], the factal paametes η, δ e τ, f paticulaly the factal exponent η dominates the phase angle esponse of the appaent complex esistivity, mainly at low fequency. Accoding to [] [], this featue is vey impotant because at low fequency the paametes cay infomation about the oughness of the poes of ocks. Thus, it becomes possible to investigate, fom 356

9 (a) (b) (c) (d) (e) Figue. Amplitude and phase angle of the appaent esistivity in a well pesenting mud and non-invasion of the fomation. The fomation is polaiable with the intinsic electical popeties given by the factal model when (a) vaying η; (b) vaying m; (c) vaying δ; (d) vaying τ and (e) vaying τf. 357

10 data of induced polaiation in the fequency domain in the well, the tanspot popeties of the geological envionment. 4.. Envionment Pesenting Thee Layes Thee thicknesses of the invaded one wee consideed when analying the induced polaiation esponse in an envionment with thee cylindical layes (mud, invaded one and fomation): one, two and five times the adius of the well. The DC esistivity of the invaded aea was pesumed equal to 0 Ω m. Figues 3-7 shows the esponse of the induced polaiation when vaying η, m, δ, τ and τf, espectively: The amplitude esponse of the appaent complex esistivity was affected by the vaiation of the invaded one. Howeve, the phase angle esponse was only slightly affected. This is simila to the esults obseved by [] [3], which demon- (a) (b) (c) Figue 3. Amplitude and phase angle of the complex appaent esistivity in a well pesenting mud, an invaded one and fomation. The invaded one and fomation ae polaiable, and the intinsic electical popeties ae given by the factal model when vaying the paamete η. The adii of the invaded one wee (a) the same; (b) twice and (c) five times the well adius. 358

11 (a) (b) (c) Figue 4. Amplitude and phase angle of the complex appaent esistivity in a well pesenting mud, an invaded one and fomation. The invaded one and fomation ae polaiable, and the intinsic electical popeties ae given by the factal model when vaying the paamete m (chageability). The adii of the invaded one wee (a) the same; (b) twice and (c) five times the well adius. states the factal natue of the complex esistivity, since the scale vaiation in the measuements did not change the phase angle esponse of the cylindical envionment. In addition, the factal exponent paamete η, which dominates the esponse phase, is not dependent on the electical popeties of the fluids filling the empty spaces of the ocks pesent in the envionment, depending only on thei minealogical composition. Thus, the influence of the invaded one is attenuated in the phase esponse. 5. Conclusion The induced polaiation esponse of a cylindical statified envionment was obtained and the factal model fo complex esistivity was applied as an intinsic electical popety of a polaiable envionment pesenting cylindical layes. The influence of the model paametes on the induced polaiation esponse was 359

12 (a) (b) (c) Figue 5. Amplitude and phase angle of the complex appaent esistivity in a well pesenting mud, an invaded one and fomation. The invaded one and fomation ae polaiable, and the intinsic electical popeties ae given by the factal model when vaying the paamete δ. The adii of the invaded one wee (a) the same; (b) twice and (c) five times the well adius. (a) 360 (b)

13 (c) Figue 6. Amplitude and phase angle of the complex appaent esistivity in a well pesenting mud, an invaded one and fomation. The invaded one and fomation ae polaiable, and the intinsic electical popeties ae given by the factal model when vaying the time constant paamete τ. The adii of the invaded one wee (a) the same; (b) twice and (c) five times the well adius. (a) (b) (c) Figue 7. Amplitude and phase angle of the complex appaent esistivity in a well pesenting mud, an invaded one and fomation. The invaded one and fomation ae polaiable, and the intinsic electical popeties ae given by the factal model when vaying the factal time constant paamete τ f. The adii of the invaded one wee (a) the same; (b) twice and (c) five times the well adius. 36

14 investigated. The esults demonstate that, as in the case of an envionment pesenting hoiontal layes, the paametes of the factal model dominate the esponse phase of the complex appaent esistivity of the envionment at low fequencies, with paticula emphasis on the factal exponent paamete ( η ), the paamete elating the esistivity of the conductive gains that block the poes of the geological envionment to the value of the DC esistivity of the ock matix ( δ ) and the time paamete of factal elaxation ( τ f ). Since the induced polaiation paametes at low fequencies cay infomation egading the oughness of the ock poes, this esult becomes impotant when intepeting petophysical data. Futhemoe, as the factal exponent depends only on the minealogical composition and the factal geomety of the envionment, the effects of the invaded one in the phase esponse ae attenuated. Additionally, the induced polaiation data makes it possible to detemine the chaacteistics of the envionment without noticeable electomagnetic coupling effects at fequencies lowe than 0 4 H. Refeences [] Da Rocha, B.R.P., Tejo, A.F., Valle, R.R.M. and Melo, M.A.B. (997) Factal Eletomagnetic Model fo Biological Systems. Pogess in Eletomagnetic Reseach,, 0-0. [] Da Rocha, B.R.P. and Habashy, T.M. (995) Factal Geomety, Poosity and Complex Resistivity I: Fom Rough Poe Intefaces to the Hand Specimens. In: Lovell, M.A., Ed., Developments in Petophysics, London Geological Society, Special Publication, London, [3] Da Rocha, B.R.P. and Habashy, T.M. (995) Factal Geomety, Poosity and Complex Resistivity II: Fom Hand Specimens to Field Data. In: Lovell, M.A., Ed., Developments in Petophysics, London Geological Society, Special Publication, London, [4] Sampaio, E.E.S., Santos, A.B. and Sato, H.K. (998) Spectal Induced Polaiation and Mineal Discimination. Pesented at the Society of Exploation of Geophysicists 68th Annual Meeting, Expanded Abstact. [5] Kemna, A., Räkes, E. and Desen, L. (999) Field Application of Complex Resistivity Tomogaphy. Pesented at the Society of Exploation of Geophysicists 69th Annual Meeting, Expanded Abstact. [6] Kemna, A., Binley, A., Ramíe, A. and Daily, W. (000) Complex Resistivity Tomogaphy fo Envionmental Applications. Chemical Engineeing Jounal, 77, [7] Debye, P. (99) Pola Molecules. The Chemical Catalog Co. Inc., New Yok. [8] Cole, K.S. and Cole, R.H. (94) Dispesion and Absoption in Dielectics. The Jounal of Chemical Physics, 9, [9] Davidson, D.W. and Cole, R.H. (95) Dielectic Relaxation in Glyceol, Popylene Glycol and n-popano. The Jounal of Chemical Physics, 9, [0] Dias, C. A. (97) Analytical Model fo a Polaiable Medium at Radio and Lowe Fequencies. Jounal de Geophysical Reseach, 77, [] Da Rocha, B.R.P. (995) Modelo factal paa esistividade complexa de ochas: Sua 36

15 intepetação petofísica e aplicação à exploação geoelética. Tese de doutoado, Cento de Geociências, Univesidade Fedeal do Paá. [] Faias, V.J.C., Maanhão, C.H.M., Rocha, B.R.P. and Andade, N.P.O. (00) Induced Polaiation fowad Modelling Using Finite Element Method and the Factal Model. Applied Mathematical Modelling, 34, [3] Faias, V.J.C., Rocha, B.R.P., Rocha, M.P.C. and Tavaes, H.R. (03) The Use of the Factal Model to Complex Resistivity in the Intepetation of Induced Polaiation Data. Applied Mathematical Modelling, 37, [4] Seigel, H. (959) Mathematical Fomulation and Type Cuves fo Induced Polaiation. Geophysics, 4, [5] Gadshteyn, I.S. and Ryhik (980) Table of Integals, Seies, and Poducts. Academic Pess, New Yok, 60 p. [6] Davis, P.J. and Rabinowit, P. (975) Methods of Numeical Integation. Academic Pess, New Yok, 459 p. [7] Rijo, L. (99) An Optimied Digital Filte fo the Cosine Tansfom. Revista Basileia de Geofísica, 0, 7-0. [8] Moan, J.H. and Gianeo, S. (979) Effects of Fomation Anisotopy on Resistivity-Logging Measuements. Geophysics, 44, [9] Feedman, R. and Vogiatis, J.P. (986) Theoy of Induced-Polaiation Logging in a Boehole. Geophysics, 5, Submit o ecommend next manuscipt to SCIRP and we will povide best sevice fo you: Accepting pe-submission inquiies though , Facebook, LinkedIn, Twitte, etc. A wide selection of jounals (inclusive of 9 subjects, moe than 00 jounals) Poviding 4-hou high-quality sevice Use-fiendly online submission system Fai and swift pee-eview system Efficient typesetting and poofeading pocedue Display of the esult of downloads and visits, as well as the numbe of cited aticles Maximum dissemination of you eseach wok Submit you manuscipt at: O contact ijg@scip.og 363