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Ultrabroadband electromagnetic wave propagation in hydrocarbon reservoirs in the presence of an oil-water interface
Russian Geology and Geophysics 50 (2009) 46–52 www.elsevier.com/locate/rgg
Ultrabroadband electromagnetic wave propagation in hydrocarbon reservoirs in the presence of an oil-water interface M.I. Epov a, *, V.L. Mironov b, S.A. Komarov b, K.V. Muzalevskii c a
Trofimuk Institute of Petroleum Geology and Geophysics, Siberian Branch of the RAS, 3 prosp. Akad. Koptyuga, Novosibirsk, 630090, Russia b Institute of Physics, Siberian Branch of the RAS, 50/38, Akademgorodok, Krasnoyarsk, 660036, Russia c Altai State University, 61 ul. Lenina, Barnaul, 656049, Russia Received 23 July 2007; accepted 27 March 2008
Abstract Numerical experiments are applied to simulate propagation of broadband pulses which are transmitted by an electric dipole and travel in a uniform oil-bearing formation or in a layered reservoir with an oil-water interface. We calculate the frequency responses of the oil- and water-saturated layers as low-frequency filters for the propagating waves and find the respective cutoff frequencies. Attenuation rates are analyzed in the cases of incident nanosecond and picosecond pulses in uniform oil-bearing formations and those reflected from the oil-water interface. The low frequency filtering of pulses is investigated in terms of its effect on wave energy attenuation during propagation in oil-bearing rocks. © 2009, IGM, Siberian Branch of the RAS. Published by Elsevier B.V. All rights reserved. Keywords: Horizontal well; fluid-saturated formation; complex permittivity; dispersive mixtures; refractive dielectric model; broadband pulse; Green’s function; simulation
Introduction A key problem in horizontal drilling is positioning the wells relative to outer and inner boundaries of the tapped oil-bearing formation (Epov et al., 1998a,b, 2002). The problem can be successfully solved using ultrabroadband or ultrashort-period pulses (Dvoretsky and Yarmakhov, 1998). In our previous paper (Epov et al., 2007) we discussed in this respect the applicability of monochromatic pulses with a Gaussian envelope and a bandwidth less than the carrier frequency, and found their attenuation to approach that of a monochromatic wave and to increase little with pulse shortening. In this study we address real ultrabroadband pulses with a bandwidth larger than the frequency corresponding to the spectrum maximum. When propagating in a dispersive medium, a pulse of this kind expands in time and attenuates by frequency dependence of the phase velocity and attenuation of frequency harmonics. The lengthening and attenuation of the pulse define the maximum sounding distance and the accuracy of positioning with respect to the oil-water interface.
* Corresponding author. E-mail address: [email protected] (M.I. Epov)
We explore the waveform changes and energy attenuation of ultrabroadband pulses that travel in a uniform oil-bearing formation and in a layered reservoir with an oil-water interface. The results we obtained may be useful in positioning horizontal wells for estimating the utmost skin depth and accuracy in sounding layered reservoirs.
Problem formulation and analytical solution Assume that in a layered hydrocarbon reservoir simulated by two halfspaces with an interface (Fig. 1) a source as a horizontal dipole is located at the height D above the oil-water interface and at some horizontal distance ∆L from the receiver (x = 0, y = ∆L, z = D). The current density at the source is given by je (x, y, z, t) = I (t)lδ(x)δ(y)δ(z − D)ex ,
(1)
where I (t) is the pulse current density, l is the dipole length, δ(x) is the Dirac delta, and ex is the unit vector of the Cartesian coordinates. Maxwell’s equations for an electric dipole in the oil-bearing halfspace are solved as in (Sommerfeld, 1947). The components of the electric field Ex are expressed as the Fourier
1068-7971/$ - see front matter D 2009, IGM, Siberian Branch of the RAS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.rgg.2008.03.010
M.I. Epov et al. / Russian Geology and Geophysics 50 (2009) 46–52
Fig. 1. Layered hydrocarbon reservoir. Transmitter and receiver are located at height z = D.
transform in the frequency and time domains and, for the time dependence e−iωt are
47
oil-bearing formation and free space, respectively, k0 = ω/c is the free space wavenumber, ω is the angular frequency of the electromagnetic wave, c is the light speed, αx and αy are the longitudinal wavenumbers, and R⊥ and R|| are the Fresnel reflection coefficients from a flat oil-water interface for TE and TM polarizations, respectively. The permittivity ε(ω) was calculated in the same way as in (Epov et al., 2007) using the refractive mixing dielectric model (Mironov et al., 2004), which Savin et al. (2007) validated in tests for an oil-saline water-sand mixture. Applying the inverse frequency-domain Fourier transform to (2) and (3) gives time dependences of broadband pulses that propagate in an oil-bearing formation to any distance from the source. In the section below we analyze the frequency response and the transfer function of plane waves in oil- and watersaturated rocks in order to gain a better understanding of attenuation and waveform change of broadband pulses in hydrocarbon reservoirs.
Exinc (xr, yr, z, ω) = ∞
dαx dαt i (α x + α y ) 2 e x r y r (k0ε(ω) 2 w 8π ωε(ω)ε0 −∞ I (ω)l
∫
−
α2x )eiw |z − D|,
(2)
Assume that the frequency response associated with propagation of a plane wave to a distance of 1 m is
Exrefl (xr, yr, z, ω) = I (ω)l 2
K(ω) = exp (−κω/c),
∞
∫ dαx dαy ei (α
8π ωε0ε(ω) −∞
Frequency response and transfer function of the two layers in a hydrocarbon reservoir
x xr
iw (z + D)
+ αy yr) e 2 2 αx + αy
2 k ε(ω) R⊥(α, ω) , × α2x R||(α, ω)w − α2y 0 w
(4)
where κω/c = Im √ ε(ω) ω/c is the attenuation rate of the harmonic component of the pulse at the response frequency and the logarithmic attenuation (3)
where Exinc is the incident field, Exrefl is the field reflected from the interface, I (ω) is the current density spectrum, w = k20ε(ω) − α2x − α2y is the transverse wavenumber, ε(ω) and √ ε0 = 8.854⋅10−12 (F/m) are the dielectric permittivities of the
Γ = (20log e) ω κ (dB), c
(5)
estimated for oil- and water-saturated layers using the dielectric model from (Epov et al., 2007) (Fig. 2, a and b, respectively). Each layer is actually a low-frequency filter (Fig. 2, a). The cutoff frequencies corresponding to the half square
Fig. 2. Frequency response (a) and logarithmic attenuation Γ (b) for oil-bearing (1) and water-saturated (2) layers.
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M.I. Epov et al. / Russian Geology and Geophysics 50 (2009) 46–52
account (see curves 6 in Fig. 3). Furthermore, with a nonzero pure conductance, the pulse peak lags some time ∆t behind that for the case when it is zero. To estimate the effect of oil grade on the frequency response of the oil-bearing layer, we compared the complex permittivity spectra for the Nurlat (Saraev et al., 2003, 2005) and North Sea (Folgero, 1995) oil fields. The greatest difference was within 1.7 and 3.6 % for the permittivity and the loss factor, respectively. Therefore, the difference being so small, the oil grade details may be safely ignored at this research stage where we are to assess approximately (in order of magnitude) the skin depth and resolution.
Changes of pulses in frequency and time
Fig. 3. Pulse transfer function for an oil-bearing formation at different distances (m): 1 — 0.5, 2 — 0.7, 3 — 1, 4 — 1.5, 5 — 2, 6 — corresponding to case of zero pure conductance when only relaxation loss is taken into account.
frequency response (dashed line in Fig. 2, b) are 14.2 MHz for the water-saturated layer and 93.6 MHz for the oil-bearing layer. Thus, high frequencies attenuate faster in the water-bearing layer than in the oil-bearing one. The logarithmic attenuation rate for high and low frequencies in the oil-bearing layer, relative to the cutoff frequency 93.6 MHz is, respectively, greater and smaller than ≈10 dB/m. Thus, attenuate mostly the high-frequency components of the spectrum. We use the frequency response of the oil-bearing layer (Fig. 2, a) to calculate the transfer function g (t) which is the response of the medium to a delta pulse. See Fig. 3 for these functions at different travel distances of a pulse plane wave. In the responses of Fig. 3, the fore fronts become progressively lower-angle and there appears an exponential tail of the propagating pulse. This tail almost disappears when pure conductance is zero and only relaxation loss is taken into
Wave propagation in a uniform oil-bearing formation. We begin with a spherical wave traveling from a dipole source in a uniform oil-bearing formation. The pulse amplitude in the transmitter was taken to be 1.0 A. See Fig. 4 for the waveforms and the respective normalized current spectra. The pulses at the level of 0.1 maximum value are 0.1 and 1.5 ns, respectively. Dipolar pulses of this kind are used, for instance, in TMG100010R01 and TMP500.080R02 transmitters manufactured by the TRIM company from St. Petersburg. The spectra were found by (2) and (3), which are the rigorous solutions of Maxwell’s equations applicable both at near-field and far-field (wave) distances from the transmitter, while (2) and (3) were calculated numerically by the Gauss formula with 24 nodes. Then FFT, with 65,536 counts, was applied to obtain the time dependence of the pulse field using the computed spectra. The computation relative error was less than 0.1 %. The spectral densities of the 1.5 and 0.1 ns pulses at far-field distances of 2.5 and 2.4 m occupy a band approximately between 0 and 0.2 GHz (Fig. 5), which is below the conventional cutoff frequency of the formation response (of the order 0.1 GHz, Fig. 2, a), while the spectra of near-field
Fig. 4. Waveforms and normalized current spectra of real broadband pulses of t = 1.5 ns (1) and t = 0.1 ns (2).
M.I. Epov et al. / Russian Geology and Geophysics 50 (2009) 46–52
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Fig. 5. Spectra of real broadband pulses of t = 1.5 ns (a) and t = 0.1 ns (b) that travel distance L in a uniform oil-bearing formation.
pulses (within 0.2 m) occupy bandwidths of 1.0–5.0 and 0.2–20.0 GHz, respectively. Thus, as far as the pulse propagates in an oil-bearing formation, its spectrum moves to the bandwidth below the cutoff frequency, and there must be some critical distance Lef at which pulse energy attenuation decreases because all monochromatic components attenuate more slowly at distances longer than Lef where the frequencies are below the cutoff (Fig. 2, b). For estimating the attenuation of the pulse energy, we consider its total energy flow P through a unit area ∞
P = ∫ |E × H|dt (W/m2).
(6)
0
This parameter is shown in Fig. 6 in a log-log scale, normalized to the total energy of a pulse that travels a distance
Fig. 6. Normalized energy flow density of pulses as a function of distance L. Regression lines 1a: P = −8.0...–62.4L, 1b: P = −46.7...–28.20L, 2a: P = −9.7... –73.7L, 2b: P = −47.1...–36.9L.
of 0.2 m in a uniform oil-bearing formation. See that the energy attenuates with distance according to a piecewise exponential law (attenuation depends weakly on distance within each curve fragment). Indeed, the attenuation rate changes (the regression lines change their slope) at some critical distance Lef about 1.7 m. See also that the shorter pulse (0.1 ns) attenuates 10 dB/m faster (a steeper regression line) than the 1.5 ns pulse because the latter has its spectrum confined to the lower-frequency bandwidth where the attenuation is lower (Fig. 2, b). Generally, attenuation is about 120 dB at distances of 2.0–2.5 m. Attenuation of waves in oil reservoirs is not the only depth limiting factor. For instance, with a dynamic range of ~120 dB in today’s radars (Reznikov et al., 2000), the skin depth would be about 1.0–1.2 m with regard to merely the energy loss. However, an increase in pulse duration because of attenuation of high-frequency components (Fig. 5) causes additional positioning errors in distances inferred from the travel time. Therefore, we consider below how the pulses change in time. Figure 7 presents computed time dependences of real 1.5 and 0.1 ns dipolar pulses of Fig. 4 that travel different distances, for the longitudinal field component Ex. All pulses are normalized to their maximum negative amplitude. The time-dependent decrease in pulse amplitude can be estimated from data of Figs. 5 and 6. As far as the pulses travel, their fronts become subject to tailing, especially the back front (Fig. 7). This behavior of the waveforms is consistent with tailing of the transfer function fronts in Fig. 3. Vainshtein (1975) wrote that the back front may tail because the energy flow signal in a lossy medium leads the loss factor, and the loss thus distorts mostly the pulse’s back front. Comparing the data in Figs. 3 and 7, one can see that the back front of the real pulse has approximately the same length as that of the transfer function at a distance of 2.0–2.5 m, whichever be the width of the transmitted pulse. However, because the latter is dipolar, its fore front at this distance (Fig. 7) is much shorter than the respective length for the transfer function (Fig. 3) corresponding to a unidirectional Dirac delta pulse. This
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Fig. 7. Waveforms of real broadband pulses of t = 1.5 ns (a) and t = 0.1 ns (b) that travel distance L in a uniform oil-bearing formation.
time-dependent behavior of the pulse agrees with its dynamics in the frequency domain (Figs. 5 and 6). The widening pulse which travels in an oil-bearing formation occupies ever greater part of the time interval between the transmission and reception points, which increases the error of its spatial positioning. We detail this problem in the next section concerning reflection of an ultrabroadband pulse from an oil-water interface. Propagation of pulses reflected from an oil-water interface. This model simulates pulse propagation in a semi-infinite oil-bearing formation that borders a water-saturated halfspace. Let the transmitter dipole and the receiver be spaced at the distance ∆L and be located at the height D above the oil-water interface (Fig. 1). See Fig. 8 for the time dependences of the real 1.5 and 0.1 ns pulses, with the longitudinal electric component calculated using (2) and (3). The transmitter-receiver separation is ∆L = 0.2 m, and Lef = √ (4D2 + ∆L2) is assumed to be the ray path length of the reflection. The pulses of Fig. 8 are normalized to their maximum absolute value as in the case of those in Fig. 7. The changes in waveforms of reflections from the oil-water interface with path lengthening are similar to the dynamics in a uniform oil-bearing formation. For estimating the spatial resolution of
soundings with pulses as in Fig. 8, we assume the distance ∆s = ∆τ⋅c/n (where n is the refractive index of an oil-bearing formation corresponding to the maximum spectral density of a ∆τ reflection) to be the maximum error in positioning the oil-water interface relative to the transmitter. The relative error in D, as plotted in Fig. 9, is the least at some optimum distance for a pulse of a given length. The accuracy is worse at distances shorter than the optimum, because the distance to be estimated is commensurate to the spatial length of the transmitted pulse, while at longer distances the cause is in pulse widening during propagation. At distances within 1 m, the positioning error is no more than 10% (Fig. 9). The dynamic potential of a broadband radar necessary to overcome attenuation in a hydrocarbon reservoir can be derived from the total energy flow of the pulse (P) reflected from the oil-water interface. Figure 10 shows the energy flow density normalized to that for a pulse that travels the distance of ∆L = 0.2 m in a uniform oil-bearing formation, according to (2) and (3), compared with the data for an incident pulse that travels the distance Lef. The attenuation rates for reflected and incident pulses are approximately the same (see the regression lines in Figs. 6 and 10), but the energy flow density in the former is on
M.I. Epov et al. / Russian Geology and Geophysics 50 (2009) 46–52
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Fig. 8. Waveforms of real broadband pulses of t = 1.5 ns (a) and t = 0.1 ns (b) reflected from oil-water interface at receiver point D(Lef) located at height D above this interface, at effective path length (Lef).
average 16 dB lower than in the latter because of loss by reflection from the oil-water interface and leakage into the water-bearing layer.
Fig. 9. Error in distance from transmitter to oil-water interface for pulses of t = 1.5 ns (1) and t = 0.1 ns (2).
Conclusions We have numerically simulated propagation of picosecond and nanosecond pulses which are transmitted by a dipole source and travel in oil-bearing formations in the absence and in the presence of reflection from an oil-water interface, and arrived at the following inferences. 1. The oil- and water-saturated layers act as low-frequency filters for partial planes waves, included in the spectrum of the dipole spherical signal, with the cutoff frequencies f0.5=93.6 MHz and f0.5=14.2 MHz, respectively. 2. The energy of a picosecond pulse (0.1 ns) traveling in a uniform oil-bearing formation attenuates faster than that of a nanosecond (1.5 ns) pulse (73.7 dB/m against 62.4 dB/m). The effect of low-frequency filtering at distances shorter and longer than 1.7 m is to make the attenuation of the 1.5 and 0.1 ns pulses about twice slower, from 62.4 to 28.24 dB/m and from 73.7 to 36.9 dB/m, respectively. 3. The attenuation rates of incident pulses and those reflected from the oil-water interface are approximately equal, while the energy flow density in the latter is on average 16 dB
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M.I. Epov et al. / Russian Geology and Geophysics 50 (2009) 46–52
Fig. 10. Energy flow density of real broadband pulses of t = 1.5 ns (a) and t = 0.1 ns (b) reflected from oil-water interface as a function of distance Lef, compared to incident pulse propagation in a uniform oil-bearing formation (Fig. 7, b). 1, 4 — uniform oil-bearing formation; 2, 5 — formation with an oil-water interface. Regression lines correspond to equations: 3a — P = −9.8... –66.8Lef; 3b — P =−65.4...–30.0Lef; 6 — P = −12.3...–72.7Lef.
lower than in the former due to loss in reflection and energy leakage into the water-saturated layer. 4. At distances within 1 m, the relative error in positioning the oil-water interface is no more than 10%. Taking into account the satisfactory amount of error in estimating the distance to the oil-water interface, the main problem in designing the respective logging tools may be assumed to consist in providing dynamic ranges of logging systems at least 120 dB for the conditions of relatively high temperatures in oil wells.
References Dvoretsky, P.M., Yarmakhov, I.G., 1998. Electromagnetic and Hydrodynamic Methods in Petroleum Prospecting and Development [in Russian]. Nedra, Moscow.
Epov, M.I., Antonov, Yu.N., Yeltsov, I.N., et al., 2002. VIKIZ Method for Logging Oil and Gas Boreholes (High-Frequency Induction Logging Isoparametric Sounding Method). Izd. SO RAN, Filial “Geo”, Novosibirsk. Epov, M.I., Mironov, V.L., Komarov, S.A., Muzalevsky, K.V. 2007. Nanosecond electromagnetic sounding of a fluid-saturated layered formation. Russian Geology and Geophysics (Geologiya i Geofizika) 48 (12), 1054–1060 (1357–1365). Epov, M.I., Sukhorukova, K.V., Nikitenko, M.N., 1998a. Estimation of parameters for thin-layered reservoirs from high-frequency inductive-logging data in horizontal boreholes. Geologiya i Geofizika (Russian Geology and Geophysics) 39 (11), 1608–1614 (1607–1614). Epov, M.I., Sukhorukova, K.V., Nikitenko, M.N., Antonov, Yu.N., 1998b. Peculiarities of high-frequency induction sounding in horizontally drilledin boreholes. Geologiya i Geofizika (Russian Geology and Geophysics) 39 (5), 649–656 (658–665). Folgero, K.A., 1995. Broad-band and high-sensitivity dielectric spectroscopy measurement system for quality determination of low-permittivity fluids. Meas. Sci. Technol. 6, 995–1008. Mironov, V.L., Dobson, M.C., Kaupp, V.H., Komarov, S.A., Kleshchenko, V.N., 2004. Generalized refractive mixing dielectric model for moist soils. IEEE Trans. Geosci. Remote Sensing 42 (4), 773–785. Reznikov, A.E., Kopeikin, V.V., Morozov, P.A., Shchekotov, A.Yu., 2000. Subsurface electromagnetic soundings: Design of instruments, processing techniques, and application experience. Usp. Fiz. Nauk, No. 5, 565–568. Saraev, D.V., Lunev, I.V., Gafarova, L.I., Yusupova, T.N., Gusev, Yu.A., Romanov, G.V. 2003. The method of dielectric spectroscopy applied to dispersion of petroleum oil. Struktura i Dinamika Molekulyarnykh Sistem, Issue 10, Part 2, 1–12. Saraev, D.V., Lunev, I.V., Yusupova, T.N., 2005. Dielectric spectroscopy for the structure of oil-bearing disperse systems. Neftegazovoe Delo, pp. 1–2. http/www.ogbus.ru Savin, I.V., Mironov, V.L., Repin, A.V., Bobrov, P.P., 2007. The effect of oil contamination on dielectric permittivity of soils, in: Proc. XI Intern. Conf. in Memory of Academician M.F. Reshetnev, Chief Designer of Rocket-Space Systems, Krasnoyarsk, 6–10 November 2007, Kranoyarsk, pp. 96–97. Sommerfeld, A., 1947. Partielle Differentialgleichungen der Physik. Vorlesungen über Theoretische Physik. Band 6. Dieterichsche Verlagsbuchhandlung (English translation, by Straus E.G., 1964: Sommerfeld, A. Partial Differential Equations in Physics. Lectures on Theoretical Physics, Volume VI, Academic Press, New York). Vainshtein, L.A. 1975. Wave Propagation [in Russian]. Izd. Ryazan Tipogr., Ryazan.
Editorial responsibility: A.D. Duchkov
Ultrabroadband electromagnetic wave propagation in hydrocarbon reservoirs in the presence of an oil-water interface M.I. Epov a, *, V.L. Mironov b, S.A. Komarov b, K.V. Muzalevskii c a
Trofimuk Institute of Petroleum Geology and Geophysics, Siberian Branch of the RAS, 3 prosp. Akad. Koptyuga, Novosibirsk, 630090, Russia b Institute of Physics, Siberian Branch of the RAS, 50/38, Akademgorodok, Krasnoyarsk, 660036, Russia c Altai State University, 61 ul. Lenina, Barnaul, 656049, Russia Received 23 July 2007; accepted 27 March 2008
Abstract Numerical experiments are applied to simulate propagation of broadband pulses which are transmitted by an electric dipole and travel in a uniform oil-bearing formation or in a layered reservoir with an oil-water interface. We calculate the frequency responses of the oil- and water-saturated layers as low-frequency filters for the propagating waves and find the respective cutoff frequencies. Attenuation rates are analyzed in the cases of incident nanosecond and picosecond pulses in uniform oil-bearing formations and those reflected from the oil-water interface. The low frequency filtering of pulses is investigated in terms of its effect on wave energy attenuation during propagation in oil-bearing rocks. © 2009, IGM, Siberian Branch of the RAS. Published by Elsevier B.V. All rights reserved. Keywords: Horizontal well; fluid-saturated formation; complex permittivity; dispersive mixtures; refractive dielectric model; broadband pulse; Green’s function; simulation
Introduction A key problem in horizontal drilling is positioning the wells relative to outer and inner boundaries of the tapped oil-bearing formation (Epov et al., 1998a,b, 2002). The problem can be successfully solved using ultrabroadband or ultrashort-period pulses (Dvoretsky and Yarmakhov, 1998). In our previous paper (Epov et al., 2007) we discussed in this respect the applicability of monochromatic pulses with a Gaussian envelope and a bandwidth less than the carrier frequency, and found their attenuation to approach that of a monochromatic wave and to increase little with pulse shortening. In this study we address real ultrabroadband pulses with a bandwidth larger than the frequency corresponding to the spectrum maximum. When propagating in a dispersive medium, a pulse of this kind expands in time and attenuates by frequency dependence of the phase velocity and attenuation of frequency harmonics. The lengthening and attenuation of the pulse define the maximum sounding distance and the accuracy of positioning with respect to the oil-water interface.
* Corresponding author. E-mail address: [email protected] (M.I. Epov)
We explore the waveform changes and energy attenuation of ultrabroadband pulses that travel in a uniform oil-bearing formation and in a layered reservoir with an oil-water interface. The results we obtained may be useful in positioning horizontal wells for estimating the utmost skin depth and accuracy in sounding layered reservoirs.
Problem formulation and analytical solution Assume that in a layered hydrocarbon reservoir simulated by two halfspaces with an interface (Fig. 1) a source as a horizontal dipole is located at the height D above the oil-water interface and at some horizontal distance ∆L from the receiver (x = 0, y = ∆L, z = D). The current density at the source is given by je (x, y, z, t) = I (t)lδ(x)δ(y)δ(z − D)ex ,
(1)
where I (t) is the pulse current density, l is the dipole length, δ(x) is the Dirac delta, and ex is the unit vector of the Cartesian coordinates. Maxwell’s equations for an electric dipole in the oil-bearing halfspace are solved as in (Sommerfeld, 1947). The components of the electric field Ex are expressed as the Fourier
1068-7971/$ - see front matter D 2009, IGM, Siberian Branch of the RAS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.rgg.2008.03.010
M.I. Epov et al. / Russian Geology and Geophysics 50 (2009) 46–52
Fig. 1. Layered hydrocarbon reservoir. Transmitter and receiver are located at height z = D.
transform in the frequency and time domains and, for the time dependence e−iωt are
47
oil-bearing formation and free space, respectively, k0 = ω/c is the free space wavenumber, ω is the angular frequency of the electromagnetic wave, c is the light speed, αx and αy are the longitudinal wavenumbers, and R⊥ and R|| are the Fresnel reflection coefficients from a flat oil-water interface for TE and TM polarizations, respectively. The permittivity ε(ω) was calculated in the same way as in (Epov et al., 2007) using the refractive mixing dielectric model (Mironov et al., 2004), which Savin et al. (2007) validated in tests for an oil-saline water-sand mixture. Applying the inverse frequency-domain Fourier transform to (2) and (3) gives time dependences of broadband pulses that propagate in an oil-bearing formation to any distance from the source. In the section below we analyze the frequency response and the transfer function of plane waves in oil- and watersaturated rocks in order to gain a better understanding of attenuation and waveform change of broadband pulses in hydrocarbon reservoirs.
Exinc (xr, yr, z, ω) = ∞
dαx dαt i (α x + α y ) 2 e x r y r (k0ε(ω) 2 w 8π ωε(ω)ε0 −∞ I (ω)l
∫
−
α2x )eiw |z − D|,
(2)
Assume that the frequency response associated with propagation of a plane wave to a distance of 1 m is
Exrefl (xr, yr, z, ω) = I (ω)l 2
K(ω) = exp (−κω/c),
∞
∫ dαx dαy ei (α
8π ωε0ε(ω) −∞
Frequency response and transfer function of the two layers in a hydrocarbon reservoir
x xr
iw (z + D)
+ αy yr) e 2 2 αx + αy
2 k ε(ω) R⊥(α, ω) , × α2x R||(α, ω)w − α2y 0 w
(4)
where κω/c = Im √ ε(ω) ω/c is the attenuation rate of the harmonic component of the pulse at the response frequency and the logarithmic attenuation (3)
where Exinc is the incident field, Exrefl is the field reflected from the interface, I (ω) is the current density spectrum, w = k20ε(ω) − α2x − α2y is the transverse wavenumber, ε(ω) and √ ε0 = 8.854⋅10−12 (F/m) are the dielectric permittivities of the
Γ = (20log e) ω κ (dB), c
(5)
estimated for oil- and water-saturated layers using the dielectric model from (Epov et al., 2007) (Fig. 2, a and b, respectively). Each layer is actually a low-frequency filter (Fig. 2, a). The cutoff frequencies corresponding to the half square
Fig. 2. Frequency response (a) and logarithmic attenuation Γ (b) for oil-bearing (1) and water-saturated (2) layers.
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M.I. Epov et al. / Russian Geology and Geophysics 50 (2009) 46–52
account (see curves 6 in Fig. 3). Furthermore, with a nonzero pure conductance, the pulse peak lags some time ∆t behind that for the case when it is zero. To estimate the effect of oil grade on the frequency response of the oil-bearing layer, we compared the complex permittivity spectra for the Nurlat (Saraev et al., 2003, 2005) and North Sea (Folgero, 1995) oil fields. The greatest difference was within 1.7 and 3.6 % for the permittivity and the loss factor, respectively. Therefore, the difference being so small, the oil grade details may be safely ignored at this research stage where we are to assess approximately (in order of magnitude) the skin depth and resolution.
Changes of pulses in frequency and time
Fig. 3. Pulse transfer function for an oil-bearing formation at different distances (m): 1 — 0.5, 2 — 0.7, 3 — 1, 4 — 1.5, 5 — 2, 6 — corresponding to case of zero pure conductance when only relaxation loss is taken into account.
frequency response (dashed line in Fig. 2, b) are 14.2 MHz for the water-saturated layer and 93.6 MHz for the oil-bearing layer. Thus, high frequencies attenuate faster in the water-bearing layer than in the oil-bearing one. The logarithmic attenuation rate for high and low frequencies in the oil-bearing layer, relative to the cutoff frequency 93.6 MHz is, respectively, greater and smaller than ≈10 dB/m. Thus, attenuate mostly the high-frequency components of the spectrum. We use the frequency response of the oil-bearing layer (Fig. 2, a) to calculate the transfer function g (t) which is the response of the medium to a delta pulse. See Fig. 3 for these functions at different travel distances of a pulse plane wave. In the responses of Fig. 3, the fore fronts become progressively lower-angle and there appears an exponential tail of the propagating pulse. This tail almost disappears when pure conductance is zero and only relaxation loss is taken into
Wave propagation in a uniform oil-bearing formation. We begin with a spherical wave traveling from a dipole source in a uniform oil-bearing formation. The pulse amplitude in the transmitter was taken to be 1.0 A. See Fig. 4 for the waveforms and the respective normalized current spectra. The pulses at the level of 0.1 maximum value are 0.1 and 1.5 ns, respectively. Dipolar pulses of this kind are used, for instance, in TMG100010R01 and TMP500.080R02 transmitters manufactured by the TRIM company from St. Petersburg. The spectra were found by (2) and (3), which are the rigorous solutions of Maxwell’s equations applicable both at near-field and far-field (wave) distances from the transmitter, while (2) and (3) were calculated numerically by the Gauss formula with 24 nodes. Then FFT, with 65,536 counts, was applied to obtain the time dependence of the pulse field using the computed spectra. The computation relative error was less than 0.1 %. The spectral densities of the 1.5 and 0.1 ns pulses at far-field distances of 2.5 and 2.4 m occupy a band approximately between 0 and 0.2 GHz (Fig. 5), which is below the conventional cutoff frequency of the formation response (of the order 0.1 GHz, Fig. 2, a), while the spectra of near-field
Fig. 4. Waveforms and normalized current spectra of real broadband pulses of t = 1.5 ns (1) and t = 0.1 ns (2).
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Fig. 5. Spectra of real broadband pulses of t = 1.5 ns (a) and t = 0.1 ns (b) that travel distance L in a uniform oil-bearing formation.
pulses (within 0.2 m) occupy bandwidths of 1.0–5.0 and 0.2–20.0 GHz, respectively. Thus, as far as the pulse propagates in an oil-bearing formation, its spectrum moves to the bandwidth below the cutoff frequency, and there must be some critical distance Lef at which pulse energy attenuation decreases because all monochromatic components attenuate more slowly at distances longer than Lef where the frequencies are below the cutoff (Fig. 2, b). For estimating the attenuation of the pulse energy, we consider its total energy flow P through a unit area ∞
P = ∫ |E × H|dt (W/m2).
(6)
0
This parameter is shown in Fig. 6 in a log-log scale, normalized to the total energy of a pulse that travels a distance
Fig. 6. Normalized energy flow density of pulses as a function of distance L. Regression lines 1a: P = −8.0...–62.4L, 1b: P = −46.7...–28.20L, 2a: P = −9.7... –73.7L, 2b: P = −47.1...–36.9L.
of 0.2 m in a uniform oil-bearing formation. See that the energy attenuates with distance according to a piecewise exponential law (attenuation depends weakly on distance within each curve fragment). Indeed, the attenuation rate changes (the regression lines change their slope) at some critical distance Lef about 1.7 m. See also that the shorter pulse (0.1 ns) attenuates 10 dB/m faster (a steeper regression line) than the 1.5 ns pulse because the latter has its spectrum confined to the lower-frequency bandwidth where the attenuation is lower (Fig. 2, b). Generally, attenuation is about 120 dB at distances of 2.0–2.5 m. Attenuation of waves in oil reservoirs is not the only depth limiting factor. For instance, with a dynamic range of ~120 dB in today’s radars (Reznikov et al., 2000), the skin depth would be about 1.0–1.2 m with regard to merely the energy loss. However, an increase in pulse duration because of attenuation of high-frequency components (Fig. 5) causes additional positioning errors in distances inferred from the travel time. Therefore, we consider below how the pulses change in time. Figure 7 presents computed time dependences of real 1.5 and 0.1 ns dipolar pulses of Fig. 4 that travel different distances, for the longitudinal field component Ex. All pulses are normalized to their maximum negative amplitude. The time-dependent decrease in pulse amplitude can be estimated from data of Figs. 5 and 6. As far as the pulses travel, their fronts become subject to tailing, especially the back front (Fig. 7). This behavior of the waveforms is consistent with tailing of the transfer function fronts in Fig. 3. Vainshtein (1975) wrote that the back front may tail because the energy flow signal in a lossy medium leads the loss factor, and the loss thus distorts mostly the pulse’s back front. Comparing the data in Figs. 3 and 7, one can see that the back front of the real pulse has approximately the same length as that of the transfer function at a distance of 2.0–2.5 m, whichever be the width of the transmitted pulse. However, because the latter is dipolar, its fore front at this distance (Fig. 7) is much shorter than the respective length for the transfer function (Fig. 3) corresponding to a unidirectional Dirac delta pulse. This
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Fig. 7. Waveforms of real broadband pulses of t = 1.5 ns (a) and t = 0.1 ns (b) that travel distance L in a uniform oil-bearing formation.
time-dependent behavior of the pulse agrees with its dynamics in the frequency domain (Figs. 5 and 6). The widening pulse which travels in an oil-bearing formation occupies ever greater part of the time interval between the transmission and reception points, which increases the error of its spatial positioning. We detail this problem in the next section concerning reflection of an ultrabroadband pulse from an oil-water interface. Propagation of pulses reflected from an oil-water interface. This model simulates pulse propagation in a semi-infinite oil-bearing formation that borders a water-saturated halfspace. Let the transmitter dipole and the receiver be spaced at the distance ∆L and be located at the height D above the oil-water interface (Fig. 1). See Fig. 8 for the time dependences of the real 1.5 and 0.1 ns pulses, with the longitudinal electric component calculated using (2) and (3). The transmitter-receiver separation is ∆L = 0.2 m, and Lef = √ (4D2 + ∆L2) is assumed to be the ray path length of the reflection. The pulses of Fig. 8 are normalized to their maximum absolute value as in the case of those in Fig. 7. The changes in waveforms of reflections from the oil-water interface with path lengthening are similar to the dynamics in a uniform oil-bearing formation. For estimating the spatial resolution of
soundings with pulses as in Fig. 8, we assume the distance ∆s = ∆τ⋅c/n (where n is the refractive index of an oil-bearing formation corresponding to the maximum spectral density of a ∆τ reflection) to be the maximum error in positioning the oil-water interface relative to the transmitter. The relative error in D, as plotted in Fig. 9, is the least at some optimum distance for a pulse of a given length. The accuracy is worse at distances shorter than the optimum, because the distance to be estimated is commensurate to the spatial length of the transmitted pulse, while at longer distances the cause is in pulse widening during propagation. At distances within 1 m, the positioning error is no more than 10% (Fig. 9). The dynamic potential of a broadband radar necessary to overcome attenuation in a hydrocarbon reservoir can be derived from the total energy flow of the pulse (P) reflected from the oil-water interface. Figure 10 shows the energy flow density normalized to that for a pulse that travels the distance of ∆L = 0.2 m in a uniform oil-bearing formation, according to (2) and (3), compared with the data for an incident pulse that travels the distance Lef. The attenuation rates for reflected and incident pulses are approximately the same (see the regression lines in Figs. 6 and 10), but the energy flow density in the former is on
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Fig. 8. Waveforms of real broadband pulses of t = 1.5 ns (a) and t = 0.1 ns (b) reflected from oil-water interface at receiver point D(Lef) located at height D above this interface, at effective path length (Lef).
average 16 dB lower than in the latter because of loss by reflection from the oil-water interface and leakage into the water-bearing layer.
Fig. 9. Error in distance from transmitter to oil-water interface for pulses of t = 1.5 ns (1) and t = 0.1 ns (2).
Conclusions We have numerically simulated propagation of picosecond and nanosecond pulses which are transmitted by a dipole source and travel in oil-bearing formations in the absence and in the presence of reflection from an oil-water interface, and arrived at the following inferences. 1. The oil- and water-saturated layers act as low-frequency filters for partial planes waves, included in the spectrum of the dipole spherical signal, with the cutoff frequencies f0.5=93.6 MHz and f0.5=14.2 MHz, respectively. 2. The energy of a picosecond pulse (0.1 ns) traveling in a uniform oil-bearing formation attenuates faster than that of a nanosecond (1.5 ns) pulse (73.7 dB/m against 62.4 dB/m). The effect of low-frequency filtering at distances shorter and longer than 1.7 m is to make the attenuation of the 1.5 and 0.1 ns pulses about twice slower, from 62.4 to 28.24 dB/m and from 73.7 to 36.9 dB/m, respectively. 3. The attenuation rates of incident pulses and those reflected from the oil-water interface are approximately equal, while the energy flow density in the latter is on average 16 dB
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Fig. 10. Energy flow density of real broadband pulses of t = 1.5 ns (a) and t = 0.1 ns (b) reflected from oil-water interface as a function of distance Lef, compared to incident pulse propagation in a uniform oil-bearing formation (Fig. 7, b). 1, 4 — uniform oil-bearing formation; 2, 5 — formation with an oil-water interface. Regression lines correspond to equations: 3a — P = −9.8... –66.8Lef; 3b — P =−65.4...–30.0Lef; 6 — P = −12.3...–72.7Lef.
lower than in the former due to loss in reflection and energy leakage into the water-saturated layer. 4. At distances within 1 m, the relative error in positioning the oil-water interface is no more than 10%. Taking into account the satisfactory amount of error in estimating the distance to the oil-water interface, the main problem in designing the respective logging tools may be assumed to consist in providing dynamic ranges of logging systems at least 120 dB for the conditions of relatively high temperatures in oil wells.
References Dvoretsky, P.M., Yarmakhov, I.G., 1998. Electromagnetic and Hydrodynamic Methods in Petroleum Prospecting and Development [in Russian]. Nedra, Moscow.
Epov, M.I., Antonov, Yu.N., Yeltsov, I.N., et al., 2002. VIKIZ Method for Logging Oil and Gas Boreholes (High-Frequency Induction Logging Isoparametric Sounding Method). Izd. SO RAN, Filial “Geo”, Novosibirsk. Epov, M.I., Mironov, V.L., Komarov, S.A., Muzalevsky, K.V. 2007. Nanosecond electromagnetic sounding of a fluid-saturated layered formation. Russian Geology and Geophysics (Geologiya i Geofizika) 48 (12), 1054–1060 (1357–1365). Epov, M.I., Sukhorukova, K.V., Nikitenko, M.N., 1998a. Estimation of parameters for thin-layered reservoirs from high-frequency inductive-logging data in horizontal boreholes. Geologiya i Geofizika (Russian Geology and Geophysics) 39 (11), 1608–1614 (1607–1614). Epov, M.I., Sukhorukova, K.V., Nikitenko, M.N., Antonov, Yu.N., 1998b. Peculiarities of high-frequency induction sounding in horizontally drilledin boreholes. Geologiya i Geofizika (Russian Geology and Geophysics) 39 (5), 649–656 (658–665). Folgero, K.A., 1995. Broad-band and high-sensitivity dielectric spectroscopy measurement system for quality determination of low-permittivity fluids. Meas. Sci. Technol. 6, 995–1008. Mironov, V.L., Dobson, M.C., Kaupp, V.H., Komarov, S.A., Kleshchenko, V.N., 2004. Generalized refractive mixing dielectric model for moist soils. IEEE Trans. Geosci. Remote Sensing 42 (4), 773–785. Reznikov, A.E., Kopeikin, V.V., Morozov, P.A., Shchekotov, A.Yu., 2000. Subsurface electromagnetic soundings: Design of instruments, processing techniques, and application experience. Usp. Fiz. Nauk, No. 5, 565–568. Saraev, D.V., Lunev, I.V., Gafarova, L.I., Yusupova, T.N., Gusev, Yu.A., Romanov, G.V. 2003. The method of dielectric spectroscopy applied to dispersion of petroleum oil. Struktura i Dinamika Molekulyarnykh Sistem, Issue 10, Part 2, 1–12. Saraev, D.V., Lunev, I.V., Yusupova, T.N., 2005. Dielectric spectroscopy for the structure of oil-bearing disperse systems. Neftegazovoe Delo, pp. 1–2. http/www.ogbus.ru Savin, I.V., Mironov, V.L., Repin, A.V., Bobrov, P.P., 2007. The effect of oil contamination on dielectric permittivity of soils, in: Proc. XI Intern. Conf. in Memory of Academician M.F. Reshetnev, Chief Designer of Rocket-Space Systems, Krasnoyarsk, 6–10 November 2007, Kranoyarsk, pp. 96–97. Sommerfeld, A., 1947. Partielle Differentialgleichungen der Physik. Vorlesungen über Theoretische Physik. Band 6. Dieterichsche Verlagsbuchhandlung (English translation, by Straus E.G., 1964: Sommerfeld, A. Partial Differential Equations in Physics. Lectures on Theoretical Physics, Volume VI, Academic Press, New York). Vainshtein, L.A. 1975. Wave Propagation [in Russian]. Izd. Ryazan Tipogr., Ryazan.
Editorial responsibility: A.D. Duchkov