Vol51,1,2008_DelNegro 159 ANNALS OF GEOPHYSICS, VOL. 51, N. 1, February 2008 Key words induced polarization – electrical dis- persion spectra – geophysical applications 1. Introduction Electric dispersion in rocks is the phenome- nology on which the Frequency-Domain (FD) Induced Polarization (IP) geophysical survey method is based. FD IP effects are mostly in- vestigated in mining, environmental and geot- hermal exploration. IP is the evidence that a complex mechanism of electrical conduction occurs in earth materials, necessitating an ex- tension of the concept of resistivity, which in the FD can no longer be retained independent of frequency. Many empirical models have been proposed to explain IP effects in rocks. Extended treat- ments can be found in the review books by Wait (1959), Bertin and Loeb (1976), Sumner (1976) and Fink et al. (1990). However, as pointed out by Wait (1982), the characterization of IP by empirical laws is vague and confusing, since, fundamentally, non-physical descriptions have been employed. To overcome this conceptual drawback, a generalized physical model has been examined in a very recent paper (Patella, 2003) by solving in the FD the following elec- trodynamic equation of a charge carrier subject to the action of an external electrical field E(ω) (1.1) where ω is the angular frequency, i is the imag- inary unit, q and m2 represent the electric charge and the mass of the carrier, m0 is an elas- tic-like coefficient accounting for recall effects, m1 is a friction-like coefficient accounting for dissipative effects due, e.g., to collisions, and R(ω) is the FT of the trajectory of the charge carrier. Assuming for simplicity that the dispersive material contains only one species of charge carriers, indicating with K the number of charge carriers per unit of volume, the following ex- pression for the dispersive conductivity func- tion σ(w), called admittivity (Stoyer, 1976), has ( ) ( ) ( )m i qR Ep p p 0 2 ω ω ω= = / Modeling electrical dispersion phenomena in Earth materials Domenico Patella Dipartimento di Fisica, Università degli Studi di Napoli «Federico II», Napoli, Italy Abstract It is illustrated that IP phenomena in rocks can be described using conductivity dispersion models deduced as solutions to a 2nd-order linear differential equation describing the motion of a charged particle immersed in an external electrical field. Five dispersion laws are discussed, namely: the non-resonant positive IP model, which leads to the classical Debye-type dispersion law and by extension to the Cole-Cole model, largely used in cur- rent practice; the non-resonant negative IP model, which allows negative chargeability values, known in metals at high frequencies, to be explained as an intrinsic physical property of earth materials in specific field cases; the resonant flat, positive or negative IP models, which can explain the presence of peak effects at specific frequen- cies superimposed on flat, positive or negative dispersion spectra. Mailing address: Prof. Domenico Patella, Dipartimen- to di Fisica, Università degli Studi di Napoli «Federico II», Napoli, Italy; e-mail: patella@na.infn.it Vol51,1,2008_DelNegro 16-02-2009 21:28 Pagina 159 160 D. Patella been derived (Patella, 2003) (1.2) Equation (1.2) is a simple IP model, which de- scribes the behavior of a circuit-like cell made of an RCL series combination. It is equivalent to the Lorentz dispersion formula obtained as solution to the 2nd-order ordinary differential equation of harmonic oscillation (Balanis, 1989). 2. Non-resonant positive IP model Let us consider now a system with two dif- ferent species of charge carriers and put with Kj, qj and mp,j (p=0,1,2) the number per unit of volume, the electric charge and the passive coefficients, respectively, of the carriers of the j-th species (j=1,2). Such a pair of ionic species can be the result of ionic dissociation of a salt dissolved in pore water. An interest- ing case to discuss is when one species (j=1) is characterized by negligible recall and iner- tia terms, and the other species (j=2) by only negligible inertia. This compound can, in fact, be used to explain the well-known IP elec- trode effect, caused by the presence of metal- lic or electronic conducting mineral particles occluding pore paths in rocks (Madden and Cantwell, 1967). Considering, for simplicity, a single-salt solution, only one ionic compo- nent, namely species 1, is assumed to undergo redox reactions at the two opposite contact faces, in order for the electric current to flow from the solution through the metallic miner- al. This is equivalent to claiming that only the passage of the reacting species 1 is allowed. The non-reactant ionic species 2, instead, af- ter running short distances under the influ- ence of the impressed field, must remain blocked along the frontal metallic face, thus giving rise to the IP effect. This simple com- pound can also explain the IP membrane ef- fect in porous rocks containing dispersed clay particles in contact with an electrolytic solu- tion (Madden and Cantwell, 1967), generally less strong than the IP electrode effect. Con- sidering again a single-salt solution, species 1 ( ) m i m m i Kq 0 1 2 2 2 σ ω ω ω ω = + - and 2 would represent, respectively, the cations, carrying the electrical current fed by the external source, and anions, traveling, in- stead, short distances due to the blocking ac- tion by the negatively charged membranes (clay particles). Referring to eq. (1.2), it is thus assumed that an elementary cell of a dispersive rock can behave like a two-branch parallel circuit, with a branch consisting of a single resistance and the other branch of a RC series sequence, i.e. (2.1) Putting (2.2a) (2.2b) (2.2c) from eq. (2.1) we obtain σ(+)(ω) as (2.3) Equation (2.3) represents a simple non-resonant positive IP model, whose real and imaginary parts are qualitatively drawn versus frequency in fig. 1. The real part of the admittivity is al- ways positive and its High-Frequency (HF) as- ymptote is located at a finite level higher than the Low-Frequency (LF) asymptote. The imag- inary part, which is always positive, vanishes at both limits and presents a maximum in the in- flexion point of the real part. The dispersion law expressed by eq. (2.3) is equivalent to the well known Debye model (Debye, 1928). Generalizing, for each elementary cell ei- ther a parallel or a series combination of N two- branch circuits can be considered. For a parallel combination, the admittivity takes the form (2.4) and for a series combination the dispersive re- sistivity function ρ(+)(ω), called impedivity ( ) ( ) i i ( ) n n n n n n N 2 1 2 1 2 1 c c σ ω ω σ ω σ σ = + + ++ = / ( ) ( ) i i ( ) 2 1 2 1 2 c c σ ω ω σ ω σ σ = + + ++ m m , , 2 1 2 0 2 c = m K q , 2 1 2 2 2 2 σ = m K q , 1 1 1 1 1 2 σ = ( ) m K q m i m i K q( ) , , ,1 1 1 1 2 0 2 1 2 2 2 2 σ ω ω ω = + + + Vol51,1,2008_DelNegro 16-02-2009 21:28 Pagina 160 161 Modeling electrical dispersion phenomena in Earth materials (Patella, 1987), takes the form (2.5) Putting (2.6a) (2.6b) (2.6c) equation (2.5) can be written in the following best known form (Patella, 2003) . (2.7) However, it is the Cole-Cole (CC) model (Cole and Cole, 1941) that has reached the greatest popularity in IP analysis since the pioneering work by Pelton et al. (1978), who reported the CC impedivity function ρCC(ω) in the form (2.8) where τ ≥ 0 and c∈[0,1] are heuristic parame- ters required to adapt eq. (2.8) to experimental data, and m∈[0,1] is the chargeability parame- ter introduced by Seigel (1959). ( ) 1 ( ) ( ) m i i 1 CC c c 0ρ ω ρ ωτ ωτ = - + < F ( ) i i 1 1 ( ) n n n N 0 1 ρ ω ρ ωτ ωβ = - + + = d n/ n n n n 0 2 1 2 2 c β ρ σ σ= n n n n n 2 1 1 2 cτ σ σ σ= + ( )lim 1 n n N 0 0 1 1 ρ ρ ω σ= ="w + = b l/ ( ) ( )i i ( ) n n n n n n N 1 2 1 2 2 1 c c ρ ω σ ω σ σ ω = + + ++ = / The CC model, though considered an em- pirical law (Wait, 1982), can physically be in- terpreted as a continuous distribution of Debye dispersion terms (Pelton et al., 1983), and hence approximated by an expression like eq. 2.7 (Patella and Di Maio, 2003). Following the theory developed by Patella (1987, 1993), the CC model has been included in the magnetotelluric (MT) method to study the distortions provoked by IP on 1D (Di Maio et al., 1991) and 2D (Mauriello et al., 1996) synthetic responses. IP effects in MT have been experimentally recognized in volcanic and ge- othermal areas (Patella et al., 1991; Coppola et al., 1993; Giammetti et al., 1996; Di Maio et al., 1997, 2000; Mauriello et al., 2000, 2004). 3. Non-resonant negative IP model Negative IP effects in earth materials are ob- served when |σ(ω)|<|σ0| for ω>0. In literature, negative IP is considered a geometric effect (Bertin and Loeb, 1976) and it was modeled for a polarizable sphere immersed in a uniform non-polarizable half-space (Sumner, 1967; Wait, 1982) and for certain layered media (Nabighian and Elliot, 1976). Sumner (1976) maintains that positive IP effects are usually as- sociated with negative IP in definite patterns, and that for most surveys, positive values are larger than negative. He warns that if these con- ditions are not seen in the data, there may well be equipment or coupling problems. Madden and Cantwell (1967) hint that negative IP ef- fects can be caused by leakages between trans- mitting and receiving circuits, and Bertin and Loeb (1976) also mention inductive coupling, which is sometimes added to and sometimes subtracted from the IP. In conclusion, negative IP in the geophysi- cal literature is not considered a physical effect, i.e. an intrinsic physical property of earth mate- rials. This is rather surprising if one considers, for instance, that negative dispersion is an in- trinsic physical property of electrons in metals at wavelengths sensibly less than about 10-2 cm (Stratton, 1941). It is shown now that a non-res- onant negative IP model can be easily derived in the frame of the theory exposed in Section 1. Fig. 1. Sketch diagram of the non-resonant positive IP model in FD. Vol51,1,2008_DelNegro 16-02-2009 21:28 Pagina 161 162 D. Patella Let us consider again a system with two dif- ferent species of charge carriers and discuss the case when both species consist of unbound charges and the first species (j=1) has negligible inertia. Referring to eq. (1.2), it is thus assumed that an elementary cell of a dispersive rock can now behave like a two-branch parallel circuit, with a branch made of a single resistance and the other branch of a RL series combination, i.e. (3.1) Using eq. (2.2a) and eq. (2.2b) and putting (3.2) eq. (3.1) is written as (3.3) Equation (3.3) represents a simple non-resonant negative IP model. The similarity with eq. (2.3) allows σ(−)(ω) to be considered a reverse Debye model. Its real and imaginary parts are qualita- tively drawn versus frequency in fig. 2. Again the real part of the admittivity is always posi- tive, but now its LF asymptote is placed at a fi- ( ) ( ) i i 1 ( ) 2 1 2 2 1σ ω ωλ σ σ ωλ σ = + + +- m m , , 2 1 2 2 2λ = ( ) m K q m i m K q( ) , , ,1 1 1 1 2 1 2 2 2 2 2 2 σ ω ω= + + - nite level higher than the HF asymptote. The imaginary part is now always negative, vanish- es again for ω→ 0, ∞ and presents a minimum in correspondence with the inflexion point of the real part. Generalizing, for each elementary cell ei- ther a parallel or a series combination of N two- branch circuits can again be considered. For a parallel combination, the admittivity takes the form (3.4) while, for a series combination, the impedivity becomes . (3.5) 4. Resonant IP models Resonant IP effects are indeed included in the general solution given by eq. (1.2), hence a system with two different species of charge car- riers is again considered by discussing the case when only the first species (j=1) is character- ized by negligible recall and inertia terms. It is thus assumed that an elementary cell of a dis- persive rock now behaves like a two-branch parallel circuit, with a branch made of a single resistance and the other branch of a RLC series combination, i.e. (4.1) which, recalling eqs.(2.2a) to (2.2c)) and eq. (3.2) can be rewritten as (4.2) Equation (4.2) represents a simple non-resonant flat IP model. Its real and imaginary parts are qualitatively drawn versus frequency in fig. 3. The real part of the admittivity is again always positive, but its LF and HF asymptotes are now placed at the same finite level (flat asymptotic ( ) ( ) ( ) i i ( )rf 2 2 2 1 2 2 2 1 2 c c σ ω ω ω λ σ ω λ ω σ σ = + - - + + ( ) m K q m i m m i K q( ) , , , , rf 1 1 1 1 2 0 2 1 2 2 2 2 2 2 2 σ ω ω ω ω = + + - ( ) ( ) i i1( ) n n n n n n N 1 2 2 1 2 1 ρ ω σ σ ωλ σ ωλ= + + +- = / ( ) ( ) i i 1 ( ) n n n n n n N 2 1 2 2 1 1 σ ω ωλ σ σ ωλ σ = + + +- = / Fig. 2. Sketch diagram of the non-resonant nega- tive IP model in FD. Vol51,1,2008_DelNegro 16-02-2009 21:28 Pagina 162 163 Modeling electrical dispersion phenomena in Earth materials line) and a bump appears with a maximum in correspondence with the resonance frequency ω=(γ2/λ2)1/2. The imaginary part goes to zero for ω→ 0, ∞ and shows an undulation with the pos- itive peak followed by the negative peak and crossing the ω-axis in the resonance frequency. It is worth pointing out that a resonant effect can be considered a sequence of a positive and a negative IP effect. Generalizing, for each elementary cell ei- ther a parallel or a series combination of N two- branch circuits can as before be considered. With no claim to be exhaustive, this survey on the IP physical modeling theory can now be concluded by considering resonance an addi- tional phenomenon superimposed on either a positive or negative IP effect. It is thus assumed that an elementary cell of a dispersive rock can now contain three different ionic species (j=1,2,3) and behave like a three-branch paral- lel circuit, with a branch made of a single resist- ance (j=1), the second branch of either a RC or a RL series combination (j=2), and the third branch by a RCL series combination (j=3), i.e. (4.3) for the resonant positive IP, and ( ) m K q m i m i K q m i m m i K q ( ) , , , , , , r 1 1 1 1 2 0 2 1 2 2 2 2 0 3 1 3 2 2 3 3 3 2 σ ω ω ω ω ω ω = + + + + + - + (4.4) for the resonant negative IP. Using the same symbolism as above, eq. (4.3) and eq. (4.4) can be given, respectively, as (4.5) (4.6) Figures 4 and 5 show qualitatively the behavior of the real and imaginary parts of the resonant positive IP model given by eq. (4.5) and the res- onant negative IP model given by eq. (4.6). The result is the sum of the diagrams of the real and imaginary parts depicted in fig. 1 and fig. 2, re- spectively, plus the homologous diagrams in fig. 3, provided that a shift to the zero-level is applied to the common LF and HF asymptote of the real part. The influence of the resonant positive IP ( ) ( ) . i i i i 1 ( )r 2 1 2 2 1 3 2 3 3 c σ ω ωλ σ σ ωλ σ ω ω λ ωσ= + + + + + - - ( ) ( ) i i i i( )r 2 1 2 1 2 3 2 3 3 c c c σ ω ω σ ω σ σ ω ω λ ωσ= + + + + + - + ( ) m K q m i m K q m i m m i K q ( ) , , , , , , r 1 1 1 1 2 1 2 2 2 2 2 2 0 3 1 3 2 2 3 3 3 2 σ ω ω ω ω ω = + + + + + - - Fig. 3. Sketch diagram of the resonant flat IP mod- el in FD. Fig. 4. Sketch diagram of the resonant positive IP model in FD. Vol51,1,2008_DelNegro 16-02-2009 21:28 Pagina 163 164 D. Patella phenomenon on transient EM methods has been studied in detail by Ageev and Svetov (1999) and Svetov and Ageev (1999). They adopted the CC model given in eq. (2.8), by arguing that for small chargeability values its validity can be ex- tended to values of c∈]1,2[ and that within this interval of c resonance effects can be modeled. 5. Conclusions It has been shown that IP phenomena in rocks can be described using conductivity dis- persion laws derived as solutions to a linear 2nd-order differential equation defining the motion of a charged particle immersed in an ex- ternal electrical field. Five dispersion laws have been discussed, namely: 1) the non-resonant positive IP model, which leads to the classical Debye dispersion law and by extension to the Cole-Cole law, largely used in current practice; 2) the non-resonant negative IP model, which introduces the possibility of explaining negative chargeability values, known in metals at high frequencies, also as an intrinsic physical effect in some earth materials; 3) the resonant flat, positive or negative IP models, which explain the presence of peak effects at specific frequencies superimposed on flat, positive or negative dispersion spectra. Though all the derived dispersion models are related to the physical parameters regulating the motion of charge carriers under the influ- ence of an external electric field, it must once again be remarked that distinguishing ionic species in earth materials by IP spectra still re- mains a difficult task. However, decrypting field measurements by one of the derived IP models may be very useful in some exploration problems. To conclude, it is worth recalling that non- resonant positive IP responses constitute by far the largest majority of effects observed directly by the IP method in mineral, oil and groundwa- ter investigations, and, indirectly by the MT method, also in geothermal areas. A wealth of literature exists on this topic, covering the last six decades almost continuously, since the pio- neering intensive works started in the late 40’s of the last century. Less known in literature are resonant IP responses, mostly studied by Russ- ian researchers, of which good examples are re- ported in Safonov et al. (1996). Finally, regard- ing the negative IP effect as a true physical property of rocks, there is not yet any documen- tation firmly attesting such a possibility. Never- theless, it is suspected that in environmental ap- plications, for instance, the observation of near- surface negative IP effects may be a useful indi- cator of the presence of massive ionic contami- nants in conductive sediments. REFERENCES AGEEV, V.V. and B.S. SVETOV (1999): The influence of rock polarizability on electromagnetic soundings, Izvestia RAS, Physics of the Solid Earth, 35, 16-24. BALANIS, C.A. (1989): Advanced Engineering Electromag- netics (J. Wiley & Sons, New York). BERTIN, J. and J. LOEB (1976): Experimental and Theoreti- cal Aspects of Induced polarization (Gebrüder-Born- träger, Berlin), 2 vols. COLE, K.S. and R.H. COLE (1941): Dispersion and Absorp- tion in dielectrics, J. Chem. Phys., 9, 341-351. COPPOLA, B., R. DI MAIO, I. MARINI, A. MERLA, D. PATEL- LA, G. PULELLI, F.M. ROSSI and A. SINISCALCHI (1993): Study of the Simplon area geothermal anomaly in the Fig. 5. Sketch diagram of the resonant negative IP model in FD. Vol51,1,2008_DelNegro 16-02-2009 21:28 Pagina 164 165 Modeling electrical dispersion phenomena in Earth materials frame of a transalpine deep railway tunnel feasibility projectin, in Underground Transportation Infrastruc- tures, Proc. of an Intern. Conf. of the Association Française des Travaux en Souterrain, edited by J.L. REITH (Balkema, Rotterdam), Toulon, 1993, 93-102. DEBYE , P. (1928): Polar Molecules (Chemical Catalogue Company, New York). DI MAIO, R., D. PATELLA and A. SINISCALCHI (1991: Sul problema del riconoscimento di uno strato elettrica- mente polarizzabile mediante misure magnetotel- luriche (in Italian), in Atti del II Convegno di Geomag- netismo ed Aeronomia, edited by A. MELONI and B. ZOLESI (Istituto Nazionale di Geofisica e Vulcanologia, Roma), 239-250. DI MAIO, R., P. MAURIELLO, D. PATELLA, Z. PETRILLO, S. PISCITELLI, A. SINISCALCHI and M. VENERUSO (1997): Self-potential, geoelectric and magnetotelluric studies in Italian active volcanic areas, Ann. Geofis., XL (2), 519-537. DI MAIO, R., D. PATELLA, Z. PETRILLO, A. SINISCALCHI, G. CECERE and P. DE MARTINO (2000): Application of elec- tric and electromagnetic methods to the study of the Phlegrean Fields caldera, Ann. Geofis., 43 (2), 375-390. FINK, J.B., E.O. MCALISTER, B.K. STERNBERG, W.G. WIEDUWILT and S.H. WARD (Editors) (1990): Induced polarization: applications and case histories, in Investi- gations in Geophysics, Society of Exploration Geo- physicists, Tulsa, Oklahoma, vol. 4, pp. 414. GIAMMETTI, S., D. PATELLA, A. SINISCALCHI and A. TRA- MACERE (1996): The Siena Graben: Combined interpre- tation of DES and MT soundings, Ann. Geofis., XXXIX (1), 189-200. MADDEN, T.R. and T. CANTWELL (1967): Induced polariza- tion, a review, in Mining Geophysics, Society of Explo- ration Geophysicists, Tulsa, Oklahoma, vol. 2, 373-400. MAURIELLO, P., D. PATELLA and A. SINISCALCHI (1996): The magnetotelluric response over two-dimensional media with resistivity frequency dispersion, Geophys. Prosp., 44, 789-818. MAURIELLO, P., D. PATELLA, Z. PETRILLO and A. SINIS- CALCHI (2000): An integrated magnetotelluric study of the Mt. Etna volcanic structure, Ann. Geofis., 43 (2), 325-342. MAURIELLO, P., D. PATELLA, Z. PETRILLO, A. SINISCALCHI, T. IULIANO and C. DEL NEGRO (2004): A geophysical study of the Mt. Etna volcanic area, in Mt. Etna: Vol- cano Laboratory, edited by A. BONACCORSO, S. CAL- VARI, M. COLTELLI, C. DEL NEGRO and S. FALSAPERLA, American Geophysical Union, Geophysical Mono- graph Series, 143, 273-291. NABIGHIAN, M.N., and C.L. ELLIOT (1976): Negative in- duced polarization effects from layered media, Geo- physics, 41, 1235-1255. PATELLA, D. (1987): Tutorial: Interpretation of magnetotel- luric measurements over an electrically dispersive one- dimensional earth, Geophys. Prosp., 35, 1-11. PATELLA, D. (1993): I principi metodologici della magne- totellurica su mezzi generalmente dispersivi (in Ital- ian), Ann. Geofis., XXXVI (5/6), 147-160. PATELLA, D. (2003): On the role of the J-E constitutive re- lationship in applied geoelectromagnetism, Ann. Geo- phys., 46, 589-597. PATELLA, D. and R. DI MAIO (1989): On the analysis of Cole-Cole relaxations transients in the induced pola- rization prospecting method, in Inverse Modeling in Exploration Geophysics, edited by A. VOGEL, R. GORENFLO, B. KUMMER and C.O. OFOEGBU (F. Vieweg & Sohn, Braunschweig/Wiesbaden), 205-219. PATELLA, D., A. TRAMACERE, R. DI MAIO and A. SINIS- CALCHI (1991): Experimental evidence of resistivity frequency-dispersion in magnetotellurics in the New- berry (Oregon), Snake River Plain (Idaho) and Campi Flegrei (Italy) volcano-geothermal areas, J. Volcanol. Geoth. Res., 48, 61-75. PELTON, W.H., S.H. WARD, P.G. HALLOF, W.R. SILL and P.H. NELSON (1978): Mineral discrimination and removal of inductive coupling with multi-frequency IP, Geo- physics, 43, 588-603. PELTON, W.H., W.R. SILL and B.D. SMITH (1983): Interpre- tation of complex resistivity and dielectric data. Part I, Geophys. Trans., 29, 297-330. SAFONOV, A.S., I.A. MUSHIN, E.S. KISELEV and A.S. GORYUNOV (1996): A structural-formation model as the physical-geological basis for high-resolution electro- prospecting, Geofizika, 2, 12-23. SEIGEL, H.O. (1959): Mathematical formulation and type curves for induced polarization, Geophysics, 24, 547- 565. STOYER, C.H. (1976): Consequences of induced polariza- tion in magnetotelluric interpretation, Pure Appl. Geo- phys., 114, 435-449. STRATTON, J. (1941): Electromagnetic Theory (McGraw- Hill, New York). SUMNER, J.S. (1967): The problem of negative IP anomalies, in Proc. of the Symposium on Induced Electrical Polar- ization, Engineering Geoscience, Department of Mineral, Technical University of California, Berkeley, 32-50. SUMNER, J.S. (1976): Principles of Induced Polarization for Geophysical Exploration (Elsevier, Amsterdam). SVETOV, B.S. and V.V. AGEEV, (1999): High resolution elec- tromagnetic methods and low fequency dispersion of rock conductivity, Ann. Geofis., 42 (4), 699-713. WAIT, J.R. (Editor) (1959): Overvoltage Research and Geo- physical Applications (Pergamon, Oxford). WAIT, J.R. (1982): Geo-electromagnetism (Academic Press, New York). Vol51,1,2008_DelNegro 16-02-2009 21:28 Pagina 165