Numerical study on scanning radiation acoustic field in formations generated from a borehole

Science in China Ser. G Physics, Mechanics & Astronomy 5 Vol.48 No. 47 56 47 Numerical study on scanning radiation acoustic field in formations generated from a borehole CHE Xiaohua 1, ZHANG Hailan 1, QIAO Wenxiao & JU Xiaodong 1. Institute of Acoustics, Chinese Academy of Sciences, Beijing 18, China;. Faculty of Natural Resources & Information Technology, University of Petroleum, Beijing 149, China Correspondence should be addressed to Che Xiaohua (email: chexiaohua@sina.com) Received December, 4 Abstract Numerical study on scanning radiation acoustic field in formations generated by linear phased array transmitters in a fluid-filled borehole is carried out using a real axis integration (RAI) method. The main lobe width of the acoustic beams and the incident angle on the borehole wall can be controlled by means of adjusting parameters, such as the element number and the delay time between the neighboring array elements of linear phased array transmitter. The steered angle of longitudinal waves generated in the formation satisfies the Snell s law for plane waves when the incident angle on the borehole wall is less than the first critical angle. When the lobe width of the acoustic beams is narrow and the steered angle is less than the first critical angle, the acoustic field in the formation can be approximately calculated given that the linear phased array is put in the formation without borehole. The technique of scanning radiation acoustic field can be applied to enhancing investigation resolution and signal-to-noise ratio in crosswell seismic survey and borehole acoustic reflection imaging. Keywords: formation, linear phased array transmitter, steered angle, RAI, scanning radiation. DOI: 1.136/144-48 The directivity of acoustic transducers used in conventional acoustic logging tools is uncontrollable [1,], which inevitably affects investigation depth and resolution. At present, deep and wide range of investigation in petroleum exploration is urgently required. It is important to improve the exploration capability to find more complex and fine reservoirs [3], for which the direction of the radiated acoustic energy is a direct factor. Acoustic field in the formations generated by the source in boreholes is involved in crosswell seismic survey and the recently developed borehole acoustic reflection imaging. The directivity and main lobe width of the acoustic beams are important properties for acoustic transmitters. Study on scanning radiation acoustic field in the formations is helpful for the interpretation for complicated near-borehole geological structures in enhancing the reliability and precision for oil and gas explorations in complex reservoirs. Qiao et al. [1,] investigated the effects of the linear phased array transmitters in Copyright by Science in China Press 5

48 Science in China Ser. G Physics, Mechanics & Astronomy 5 Vol.48 No. 47 56 fluid-filled boreholes. However, they emphasized the analysis of the acoustic field within the boreholes. Dong et al. [4,5] established a general algorithm for modeling source radiation from open and cased boreholes in layered transversely isotropic (TI) media and cylindrical isotropic media by using a boundary element method (BEM), but the directivity of the sources used in the analysis was omnidirectional and the sources were in a low frequency range. Che et al. [6] had a study on the acoustic field in formations generated by several kinds of sources in boreholes, as well as the method to control the direction of the radiated energy to the formations using the finite element method (FEM). Because of a limited computer capability, detailed study on the properties of radiation acoustic field was not given, and only qualitative results were presented. Zhang et al. [7 9] studied borehole acoustics using the RAI method, but did not investigate the acoustic field in the formations generated by phased array transmitters. Although it is simple compared to some methods [6,1,11], such as the FEM and the finite difference method (FDM), the RAI [7,8] method can save much computation time and give precise results. How to generate scanning radiation acoustic field in formations, and the relationship between the steered angles of the acoustic beams of the linear phased array transmitters in the boreholes and the generated acoustic field are quantitatively studied using the RAI method in this paper. 1 Theory 1.1 Scheme of numerical modeling As shown in fig. 1, in the cylindrical coordinate system (r,θ, z), there is a borehole with the radius of a which is infinite in z direction. The borehole axis is aligned with the z-axis, while the r-axis is along the borehole radial direction. The borehole is filled with fluid, and its compressional (P) wave velocity v f and density are 15 m/s and 1 kg/m 3, respectively. The formation is infinite and homogeneous, with velocities of compressional and shear waves v P and v S, and density being 45 m/s, 5 m/s, and 3 kg/m 3, respectively. An m-element linear phased array transmitter T is located on the borehole axis. Each element is a cylindrical vibrator with radius r and height h. The elements are excited one by one from the first element with a time delay τ. Both the center of the transmitter T and the point O are on z =. Wave field is recorded by a circular array consisting of 361 receivers, each receiver is 3. m from Fig. 1. Scheme of numerical modeling. O. The angle between the jth receiver-o line Copyright by Science in China Press 5

Numerical study on scanning radiation acoustic field in formations generated from a borehole 49 and the r-axis is denoted as α j ( 9 < α j < 9 ), which satisfies α j α j +1 =.5. 1. Sources and F-K spectrum According to the model parameters, the first critical angle of the acoustic wave incidence on the borehole wall is 19.47. In order to investigate the relationship between the steered angles of the acoustic beams generated by the linear phased array transmitters and the acoustic field, the main lobe width should be as narrow as possible. In addition, side lobe of the acoustic beams should be weak. The vertical directivity [1] of a 14-element linear phased array T is located as shown in fig., and the center frequency of the source is 1. khz. Each element is simplified as a point source in the calculation, and the inter-element spacing is 6. cm. The delay time of the exciting signals exerted on the neighboring array elements is respectively. µs, 5. µs and 13.333 µs, which generates the corresponding acoustic beams with steered angle of. 7.18 and 19.47, respectively. Fig.. Directivity of a linear phased array transmitter. The driving signal is sine wave modulated by a gauss envelope. When the parameters of the linear phased array, such as the element number, the distance and the delay time exerted on the neighboring elements, are changed, the energy of the compressional waves or the shear waves can be enhanced or weakened. The displacement named ur (,) z t on the surface of the transmitter is u r (,) z t u ( t+ ( l 1) τ ), z < mh /, l = 1,,, m, =, z > mh/, www.scichina.com

5 Science in China Ser. G Physics, Mechanics & Astronomy 5 Vol.48 No. 47 56 so the F-K spectrum is 1.3 Theoretical analysis 1 U ( k, ω) = u ( z, t)exp( ikz+ i ωt) dzdt. () r r π Acoustic field in a fluid-filled borehole satisfies the acoustic equation with only one scalar function ϕ f, that is, f ( ) ( ) ϕ ( rk,, ω) = Dk (, ω) H α r + Ak (, ω) J αf r, (3) f where the radial wave number α f satisfies ω f k vf α =. Acoustic field in formations satisfies elastic equations. Potential functions ϕ and η are used to characterize the displacements and the stresses in the formations, ( ) ϕ(, rk, ω) = Bk (, ω) H αr, (4) ( ) η(, rk, ω) = Ck (, ω) H β r. (5) The potential functions ϕ and η satisfy the wave equations for the compressional waves and the shear waves, and ω k vp α =, ω β = k are radial wave num- v bers, respectively, and B and C are unknown parameters related with k and ω that can be determined by boundary and source conditions. According to the boundary conditions on the borehole wall r = a and on the source surface r = r, the potential functions ϕ f (r,k,ω), ϕ (r,k,ω) and η (r,k,ω) can be written as S H ( αfr) + E( k, ω) J( αfr) (,, ) (, ), [ H ( r ) E( k, ) J ( r )] ϕf rkω = Ur kω α α + ω α f 1 f 1 f (6) Fk (, ω) H ( αr) (,, ) = (, ), [ H ( r ) E( k, ) J ( r )] ϕ rkω Ur kω α α + ω α f 1 f 1 f ( β ) Gk (, ω) H r (,, ) (, ). [ H ( r ) E( k, ) J ( r )] η rkω = Ur kω α α + ω α f 1 f 1 f (7) (8) The coefficients of E, F and G in eqs. (6) (8) are similar to the coefficients of A, B and C in eqs. (3.) (3.4) in ref. [7]. Substituting eqs. (6) (8) into the integration such as Copyright by Science in China Press 5

Numerical study on scanning radiation acoustic field in formations generated from a borehole 51 the potentials can be obtained. Results and analysis 1 ϕ(,,) rzt = ϕ(,, )exp(i i ), rk ω kz ω tdkd ω π (9) The direction and lobe width of the acoustic beams cannot be controlled if a source only consists of one element. In this case, the direction of acoustic propagation in the formations cannot be steered. It is possible to make the lobe width narrow, which contributes to higher detection resolution and higher signal-to-noise ratio if multi-element linear phased array is adopted and designed properly. Even more, the capability for detecting complex geological structures will be enhanced in the case of applying the scanning radiation acoustic field technique..1 Acoustic field in the formations generated by linear phased arrays Fig. 3(a) (e) show acoustic waveforms generated by a 14-element linear phased array transmitter, with the height of 6. cm, and diameter of 5. cm, for each cylindrical element. The diameter of the fluid-filled borehole is. cm, and the delay time between the neighboring elements is respectively., 5., 1., 13.333 and 4. µs. The delay time τ =13.333 µs corresponds to the steered angle close to the first critical angle, and τ =4. µs corresponds to the steered angle close to the second critical angle. As shown in fig. 3(a), acoustic waves generated in the formation are symmetrical with α j =. when τ equals. µs. The amplitude of the compressional waves slightly decreases with the increasing of α j, and the energy of the compressional waves is obviously weak when α j > 5.. With the increasing of τ, as shown in fig. 3(b) (d), the compressional waves generated in the formation clearly steer to the area of α j >., while the energy radiated to the area of α j <. is much weaker. In addition, the steered angle of the compressional waves increases with the increasing of τ. When the steered angle of the main radiation lobe is less than the first critical angle, the acoustic energy can radiate from the fluid-filled borehole into the formation. The energy of the compressional waves is rather weak when τ equals 4. µs, whereas the shear energy is very strong, as shown in fig. 3(e). It is found that the acoustic energy radiates into a relatively narrow range by the method of making the main lobe width of the linear phased array narrow. The steered angle can be controlled by adjusting the delay time exerting on the neighboring elements of the linear phased array, so the radiation direction of the acoustic energy from the fluid-filled borehole to the formation. Scanning radiation acoustic field to different directions in the field can be achieved by the method of point measurement in the borehole, that is, makes the incidence on the borehole wall less than the first critical angle and continuously increases or decreases the delay time exerting on the neighboring elements of the linear phased array. www.scichina.com

5 Science in China Ser. G Physics, Mechanics & Astronomy 5 Vol.48 No. 47 56 Fig. 3. Waveforms generated by a linear phased array transmitter. (a)τ =. µs; (b) τ = 5. µs; (c) τ = 1. µs; (d) τ = 13.333 µs; (e) τ = 4. µs. Fig. 4 demonstrates the amplitude distribution of the compressional waves with an angle of α j when τ equals., 4., 8., 1. and 13.333 µs, respectively. With the increasing of the delay time exerting on the neighboring elements of the linear phased array, the amplitude of the compressional waves in the formation decreases correspond- Copyright by Science in China Press 5

Numerical study on scanning radiation acoustic field in formations generated from a borehole 53 ingly. However, the steered angle increases gradually. The lobe width of the acoustic beams generated by the linear phased array should be as narrow as possible to make more energy into the formation in crosswell seismic survey and borehole acoustic reflection imaging. Fig. 4. Distribution of the compressional waves in formation generated by the transmitter in a borehole. The angle of refraction θ t can be calculated according to Snell s law and incidence on the borehole wall for different delay time of the linear phased array transmitter from. µs 13.333 µs. Fig. 5 shows the relationship between θ t, θ max and the delay time of the linear phased array transmitter, where θ max stands for the steered angle. It is clear that θ t has a close agreement with θ max when τ < 8. µs, and there is a little difference when τ > 9. µs.. Borehole effects In order to take effects of a borehole on the acoustic field generated in formations into account, the borehole in fig. 1 is supposed to be neglected, that is, the model is redefined as homogeneous infinite formation without borehole, and the cylindrical elements are replaced by point sources. It is found that the amplitude of the compressional waves for without borehole case is much stronger than with a borehole case. The amplitude of the compressional waves for both with the borehole and without borehole cases is normalized by their corresponding maximum amplitudes and compared with the vertical directivity of the linear phased array, as shown in fig. 6(a) (c) with τ =. µs, τ = 8. µs, and τ = 13.333 µs, respectively. The three curves show a good consistency for each τ. So, when only the acoustic field in the homogeneous formations generated by the linear phased array transmitters in the borehole is considered, it is proper to suppose www.scichina.com

54 Science in China Ser. G Physics, Mechanics & Astronomy 5 Vol.48 No. 47 56 Fig. 5. The relationship between θ t, θ max and the delay time of the linear phased array transmitter. that the linear phased array transmitter is put in the formation without borehole, and the acoustic field can be calculated from the directivity function for the solid formation. This method is similar to the mirror image method, which is widely used for studying static electric field. Therefore, we name it an acoustic mirror image method. If the lobe width of the acoustic beams is narrow, and the steered angle is less than the first critical angle, the acoustic mirror image method makes the numerical modeling for scanning radiation acoustic field simple and easy greatly. 3 Conclusions (i) Numerical studies on scanning radiation acoustic field in formations generated by linear phased array transmitters in a fluid-filled borehole is presented by using a RAI method. The lobe width of acoustic beams can be controlled by adjusting parameters, such as the element number and the inter-element spacing, of the linear phased array transmitters, while the steered angle can be changed by adjusting the delay time exerting on the neighboring elements. The radiation energy to certain direction is enhanced with these treatments. When the incident angle on the borehole wall is less than the first critical angle, the steered angle of longitudinal waves generated in the formations satisfies the Snell s law for plane waves. So, scanning radiation of acoustic field to the formations can be achieved through adjusting the delay time exerting on the neighboring elements of the linear phased array transmitters. (ii) If the lobe width of the acoustic beams is narrow, and the steered angle is less than the first critical angle, the acoustic field in formations can be obtained by the acoustic mirror image method, which is a great simplification for numerical calculation. The Copyright by Science in China Press 5

Numerical study on scanning radiation acoustic field in formations generated from a borehole 55 Fig. 6. The directivity curve and the distribution of the compressional waves in formations generated by the linear phased array transmitter for the cases with a borehole and without borehole. (a)τ =. µs; (b) τ = 8. µs; (c) τ = 13.333 µs. method supposes that the linear phased array is put in the formations without borehole, and uses the directivity function to approximatively calculate the acoustic field in the formations. (iii) At present, new generation acoustic technology is an urgent requirement in petroleum exploration, for enhancing the capability to find more complex and fine oil reservoirs. In the crosswell seismic survey and borehole acoustic reflection imaging, scanning radiation acoustic field technique will be helpful for the interpretation of complicated oil and gas reservoirs, for great investigation depth and wide range of investigation and for an improvement of data quality. It is possible to apply the scanning radiation technique to petroleum exploration through the point measurement method. The assembly technique can be applied in acoustic logging, borehole acoustic reflection imaging, as well as crosswell seismic survey. (iv) In order to apply this technique to petroleum industry, appropriate electric circuits are needed to drive and control multi-element linear phased arrays which work in the circumstances with a high temperature, a high pressure and a narrow space. Acknowledgements The first author wishes to thank Prof. Wang Xiuming in CSIRO Petroleum of Australia for revising this paper. This work was supported by the National Natural Science Foundation of China (Grant Nos. www.scichina.com

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