Full-wave synthetic acoustic logs in porous media

Transcription

1 Global Geology doi /j. issn Article ID Full-wave synthetic acoustic logs in porous media LI Hanqing WANG Zhuwen and ZHANG Xueang College of Geo-Exploration Science and Technology Jilin University Changchun China Abstract In order to investigate the permeability of porous media it is necessary to figure out the property of wave propagation in this medium. The authors calculated the full waveforms and the 2-dimensional spectrum of fluid saturated porous media analyzed the wave propagation characteristics of different permeable stratums. From the theoretical acoustic pressure waveform received at the well-axis and the 2-D spectrum it is revealed that longitudinal- transverse- Stoneley- pseudo-rayleigh- longitudinal mode- and transverse mode waves are different in characters in various permeable strata. Stoneley wave is sensitive to the change of permeability and its frequency range and amplitude attenuation are influenced by permeability. Key words acoustic logging full wave-form porous media numerical calculation 1 Introduction Full waveform in acoustic logging is a major breakthrough in acoustic logging. It makes a regular record of the single longitudinal wave arriving time extend to the records of the longitudinal wave sliding transverse wave and the guided wave in the fluid between the borehole wall and the logging instrument which contain large quantities of information of the strata. In order to obtain this stratigraphic information we must study the characteristics of wave propagation in strata with different properties. As for porous media Biot theory about the acoustic field propagation in pore space media has been widely recognized and applied in depicting pore space mediums at present which predicted that there are two kinds of longitudinal waves-fast and slow longitudinal wave. This prediction was proved by Plona 1980 with experiment data. Porous space media theory of Biot which we called Biot- Roserbaum theory was applied in the research of acoustic logging by Rosenbaum 1974 for the first time. The influence of reservoir stratum parameters such as porosity and permeability was studied numerically with point source model and then he pointed out that permeability and Stoneley wave had the closest relation. In 1987 Johnson introduced the concept of dynamic permeability and applied it in Biot theory and then he got an approximate formula of dynamic permeability Schmitt obtained the 2-dimensional spectrum of elastic media. 2 Full-wave synthetic acoustic logs in porous media Elastic wave propagation in porous media is a complex issue among the different research methods Biot theory has been used and recognized widely. Biot theory of porous media model made some assumptions and obtained stress-strain relations and the equations of motion in porous media. According to Biot theory total stress component is Received 5 January 2012 accepted 10 February 2012

2 152 Li H. Q. Wang Z. W. and Zhang X. A. σ ij = 2μe ij + λe - αp δ ij 1 The pressure in the fluid is p = - α u + w /β Dynamic Darcy's law is 2 w = v - u = θ p - ρ pf ω 2 u 3 By substituting expressions in 2 to 1 and 3 we obtain λ + μ u + μ 2 u + ρ ~ ω 2 u - α ~ p = 0 Where θ 2 p - βp - α ~ u 4 5 λ = 1 - λ s μ = 1 - φ μ s α = 1 - K b /K s K b = 1 - λ s + 2μ s /3 ρ = ρ s ρ pf θ = iκ ω /ηω β = α - /K s + /K pf e = u v and u are the displacement vectors of porous solid and fluid respectively μ is the bulk strain of porous solid K s λ s μ s are the bulk modulus and Lame coefficients of solid skeleton ρ s is the density of solid skeleton ρ pf is the density of the fluid in porous media and η is viscosity coefficient. In the cylindrical coordinate system r θ z let us assume that the borehole axis coincides with the Z axis R axis stands for radius direction axisymmetric model. In the borehole it is fluid filled and surrounded by a porous medium outside. Let us assume the spectral function of the sound is S ω we can obtain the total pressure p r z t in the borehole. It is the sum of the source term and the scattered wavefield and it can be presented in the following form p r z t = 1 S ω e jωt K 4π 2 - dω 0 ξ f r + - A' ω k 2 I 0 ξ f r e -jk z z dk z 6 Where ξ f = k 2 z - k 2 f 1 2 kf = ω /v f A' ω k 2 is formation filter function and can be given by matrix division. The matrixes are functions of permeability. In 1987 Johnson introduced the concept of the dynamic permeability and gave an approximate formula of dynamic permeability κ ω = κ ( ) 2 - iωα -1 ρ f κ [ ] 1 - i4α2 κ 2 ρ f ω Λ 2 2 η η 7 8α Where Λ = κ 0 槡 α = /2. In above equations is the porosity k 0 is static permeability. 3 Results of full waveform processing In order to calculate the integral in formula 6 we used an adaptive integration procedure for the expressions containing trigonometric functions. The calculation of the integral over frequency was performed by applying a fast Fourier transform. We simulated the full waveform in porous media. The formation properties and parameters of the acoustic tools and borehole are given in Table 1. In this paper the monopole sound source we use is a kind of cosine wave packet pluse. Its radius equals to m. T c is pluse length f 0 is source center frequency and ω 0 is source center angular frequency. In the frequency domain it can be expressed as S ω = 0.25T c sin 0.5T cω + 0.5T c ω 0 + sin 0.5T cω - 0.5T c ω T c ω + 0.5T c ω 0 0.5T c ω - 0.5T c ω sin 0.5T cω + 0.5T c ω 0 + π 0.5T c ω + 0.5T c ω 0 + π + sin 0.5T cω - 0.5T c ω 0 - π + 0.5T c ω - 0.5T c ω 0 - π sin 0.5T c ω + 0.5T c ω 0 - π + sin 0.5T cω - 0.5T c ω 0 + π 0.5T c ω + 0.5T c ω 0 - π 0.5T c ω - 0.5T c ω 0 + π e0.5t cω Table 1 V sp Parameters of the stratum and the source V ss V bp V bs V f v B ρ s ρ f ρ B η a -1 /kg m -1 s /m f 0 /khz 3. 1 Permeability's influence on frequency dispersion By solving the functions in formulas 4 and 5 we can obtain velocities of fast and slow longitudinal wave which changes with frequency. To study the influence of permeability on the velocity we varied permeability from 1. 0 to m 2 and graphed the result in Fig. 1.

3 Full-wave synthetic acoustic logs in porous media 153 As we can see in Fig. 1 with the increase of permeability the fast wave velocity changes slower and slower. The velocity has bigger value and slower increasing rate in high permeability than in low permeability. The velocity changed 1. 95% and 0. 22% in 1. 0 and m 2 from 1 ~ 30 khz. Fig. 1 Frequency dispersion curves of porous formation with different permeability 3. 2 Acoustic pressure waveform at the well-axis Fig. 2 shows the pressure waveforms at the wellaxis of porous stratum with different permeability properties. The permeability is m 2 and source spacing is 2 m. We can see clearly that there are P-wave S-wave and Stoneley wave in the borehole. Stoneley wave has a lower velocity than that of mud wave. In this figure it arrives at 2. 2 ms. As can be seen from the graph with the increase of permeability the wave amplitude reduced after 2. 2 ms. Stoneley wave is sensitive about the change of permeability. The greater permeability changes the faster amplitude decays Two-dimensional spectrum of fluid saturated porous media To analyze the characters of the partial wave we applied 2-D spectrum numerical value in frequency - wave number domain. We know that formation filter function A' ω k z has a relation with the boundary conditions and the properties of the surrounding stratum. The numerical value of A' ω k z in frequencywave number domain is called two-dimensional spectrum. Two-dimensional spectrum reflects the filter quality of the stratum. We calculated the value of 2-D spectrum of porous stratums with different permeability and figured it in frequency-wave number domain Fig. 3. The right axis denotes the frequency the left axis denotes the wave number and the gray scale represent the value. In the 3 graphs we can see three indicated thin diagonals. From left to right diagonal signifies the fluid wave S-wave of the stratum and P-wave of the stratum respectively group velocity. Since the valve of 2-D spectrum near the diagonals denote the amplitude of the wave at the borehole and the scales there are low we can know that the amplitudes of all three waves are very small. Because of dω /dk z = ω /k z phase velocity equals to group velocity the phase velocity and the group velocity of the three waves change rarely in borehole acoustic field. In other words they have the character of weak frequency dispersion. In Fig. 3 close to diagonal 1 we can see the 2-D spectrum of Stoneley wave which is the contribution of the real-poles in the interval of k 2 z > ω 2 /v 2 f. It is a kind of transmission mode. The dark stripes between diagonal 1 and 2 are the transmission modes of pseudo-rayleigh wave. They are the contribution of the real-poles in the interval of ω 2 /v 2 f ω 2 /v 2 s. The dark stripes between diagonal 2 and 3 are the leakage mode waves of S -wave. They are the contribution of the complex-poles. The leakage mode wave is a kind of geometric attenuation wave and shows the characteristics of short propagation path high velocity and fast attenuation. The dark stripes between diagonal 3 and the abscissa are the leakage mode waves of P-wave. They are also the contribution of the complex-poles and have the same properties with the leakage mode waves of S-wave. Along with the increase of permeability the dark stripes near diagonal 1 gradually diminished. This means that Stoneley wave is sensitive about the change of permeability. The speed of the Stoneley wave lower than that of fluid wave in the borehole and the frequency dispersion is weak. The frequency range is small and has cut-off frequency so the range of fre-

4 154 Li H. Q. Wang Z. W. and Zhang X. A. Fig. 2 Acoustic pressure waveforms of porous formation with different permeability quency is 0 to cut-off frequency. When permeability increasing the amplitude of Stoneley wave attenuates more intensely. The cutoff frequency of Stoneley becomes smaller with the increase of permeability and has a smaller range. 4 Conclusions In this paper we studied the wave propagating mode in a cylindrical borehole embedded in fluid saturated stratum. In the process we introduced the concept dynamic permeability into Biot-Rosenbaum and calculated the 2-dimensional spectrum. From the acoustic pressure waveform and the 2-D spectrum we can see that Stoneley wave is sensitive about the change of permeability. The frequency range and amplitude attenuation of Stoneley wave are effected by permeability.

5 Full-wave synthetic acoustic logs in porous media 155 a K = m 2 b K = m 2 c K = m 2 Fig. 3 2-D spectrum of porous formation with different permeability References Biot M A Theory of propagation of elastic waves in a fluid-saturated porous solid I-low-frequency range. J. Acoust. Soc. Am Biot M A Theory of propagation of elastic waves in a fluid-saturated porous solid. II-higher frequency range. J. Acoust. Soc. Am Hu H S Wang K X Dynamic permeability in porous medium acoustic. Chinese Journal of Geophysics in Chinese with English abstract Johnson D L Koplik J Dashen R Theory of dynamic permeability and tortuosity in fluid-saturated porous media. Fluid Mech Johnson D L Plona T J Kojima H Probing porous media with first and second sound. Ⅱ. Acoustic properties of water-saturated porous media. J. Appl. Phys Plona T J Observation of a second bulk compressional wave in a porous medium at ultrasonic frequencies. Appl. Phys. Lett Rosenbaum J H Synthetic microseismograms logging in porous formations. Geophysics Schmitt D P Acoustic multipole logging in transversely isotropic poroelastic formations. Acoust. Soc. J. Am Tang X M Cheng C H Toks z M N Dynamic permeability and borehole Stoneley waves a simplified Biot- Rosenbaum model. Acoust. J. Soc. Am Yue C W Wang Z W Chen B T Analysis on acoustic logging response using two-dimensional spectrum with the change of the diameter. Progress in Geophysics in Chinese with English abstract Yue C W Wang Z W Calculation of the borehole a- coustic field based on boit-rosenbaum theroy and dynamic permeability. Progress in Geophysics in Chinese with English abstract