2D FINITE-ELEMENT NUMERICAL SIMULATION OF SONIC MEASUREMENTS ACQUIRED IN THE PRESENCE OF A MANDREL TOOL

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1 D FINITE-ELEMENT NUMERICAL SIMULATION OF SONIC MEASUREMENTS ACQUIRED IN THE PRESENCE OF A MANDREL TOOL Paweł J. Matuzyk 1, Lezek Demkowicz 1, David Pardo, Jun Ma 1, and Carlo Torre-Verdín 1, 1 The Univerity of Texa at Autin, Baque Center for Applied Mathematic, Bilbao, Spain ABSTRACT We preent a method for the imulation of onic logging meaurement uing frequency-domain-baed computation. A limited number of frequencie (typically below 50) are ufficient to accurately reproduce diperion curve, which are ued for the aement of oil- and ga-bearing formation. We ue an hp-fem refinement trategy to perform highly-accurate frequency domain imulation. Verification reult are diplayed, both in term of the frequency pectrum a well a in term of diperion curve. Numerical reult illutrate the flexibility of the method, which can be employed to ae formation containing fracture and/or thin and layer, and can be readily extended to the cae of borehole-eccentered tool and/or deviated well. INTRODUCTION Acoutic logging ha been extenively ued by oil companie to meaure formation material propertie uch a poroity and mechanical rigidity. In order to undertand and tudy the acquiition of onic logging meaurement, a plethora of numerical method have been developed, including emi-analytical method [1,, 3, 11], and timedomain finite difference [4, 5, 6]. Numerical imulation of onic logging meaurement have been almot excluively performed in the time-domain (a few exception can be found in the literature, e.g, [7, 8]). Furthermore, frequency-domain baed method have lacked competitivene with repect to time-domain method, ince a large number of frequencie need to be computed in order to obtain an accurate time repone. Figure 1 and diplay two typical pectra of onic logging meaurement, and illutrate the rapid variation that occur a we modify the frequency. Thee rapid variation in the amplitude of the preure a a function of frequency demand the ue of a large number of denely located frequencie (over 500) in order to perform an accurate invere (fat) Fourier tranform. We propoe a new approach for performing frequency-domain baed imulation of onic logging meaurement. The main idea of thi paper i to obtain directly diperion curve from frequency domain reult, without ever obtaining a time-domain ignal. After all, diperion curve already contain the information about the lowne of the formation, which i employed by petro-phyicit to obtain phyical propertie of the reervoir. Since diperion curve are mooth with repect to variation in frequency (ee Fig. 3), it i enough to calculate reult only for a limited number of frequencie (below 50) to accurately reproduce diperion curve. In addition, we believe that it i poible to further reduce the number of computed frequencie to a number below 15 when only V p (high frequencie) and V (low frequencie) are needed. Furthermore, in the computer-aided imulation there i neither need to utilize a Ricker wavelet nor any other wavelet, becaue for each frequency, we only need information about the repone of the onic tool for a ource with an arbitrary magnitude. Other advantage of uing a frequencydomain-baed approach include: (a) the added tability achieved when imulating the problem, ince there i no need to impoe mall time-tep to aure tability of the problem, (b) the uperior performance achieved by Perfectly Matched Layer (PML ) in frequency domain computation (there i no need to worry about cauality), and (c) the maller computational complexity of the imulation problem aociated to frequency-domain-baed computation. On the other hand, the main challenge of frequency-domain-baed imulation of onic logging meaurement include: (1) the multi-phyic nature of the problem (coupled acoutic and elaticity), () the truncation of the computational domain, which involve the ue of PML, (3) the need to properly treat different ource uch a monopole, dipole, and quadruple, and (4) the o-called diperion error aociated to the imulation of highfrequency problem, which produce a delay of the traveling wave in the numerical olution. We note that the firt three challenge are common to the ue of time-domain-baed imulation.

2 Matuzyk et al. METHOD In order to overcome the above challenge, we propoe the ue of a fully automatic hp-adaptive multi-phyic finite element method (hp-fem). Thi oftware incorporate a multi-phyic automatic hp adaptive algorithm, which optimally adapt the meh for each frequency, both with repect to element ize h and polynomial order of approximation p, delivering exponential convergence rate in all imulation problem. In other word, thi method deliver highly accurate olution uing a minimal number of degree of freedom in comparion to other method that employ other type of adaptation (h, p or r). The adaptive algorithm employ in each tep a two-grid trategy (coare and fine meh) to etimate the error of the olution, which i utilized to make deciion about optimal meh refinement. Thi hp-fem oftware i utilized to accurately and efficiently imulate borehole acoutic meaurement. The ue of high-order method (p>1) dratically reduce the diperion error [9], which enable one to obtain accurate imulation reult. In addition, the hp-fem i ideally uited to olve boundary layer that necearily arie from the ue of PML [8, 10], which we employ to truncate the computational FE domain. In thi work, we have developed a new verion of the hp-fem oftware enabling modeling of phyical phenomena that i decribed by a et of coupled partial differential equation (PDE ). In the cae of onic logging meaurement, we have two different domain: (a) the acoutic domain, compoed of the borehole fluid, and (b) the elatic domain, compoed by the tool, formation, and poibly caing. FORMULATION Mathematically, we olve the following et of coupled PDE in the frequency domain (quantitie p, u, σ are the Fourier tranform of the preure, diplacement, and tre tenor, repectively): p c p 0 f u 0 n p n u T I u u u f f n pn where c f i the fluid peed ound, ω i the frequency, ρ f and ρ are the fluid and olid denitie, repectively, n f and V V and n are the outide normal vector for the acoutic and elatic domain, repectively, p V are the o-called Lamé contant (V p and V are the P- and S- wave velocitie in the elatic olid, repectively). In the above et of equation, the firt one decribe the preure phenomena in the acoutic domain, the econd one i a momentum equation for the olid (linear elaticity), the third one i a contitutive equation, while the lat two equation decribe the coupling exiting between both the acoutic and elatic domain. NUMERICAL RESULTS: The new code ha been compared againt everal alternative numerical imulation method uch a a 1D emianalytical code developed by Jun Ma [11], and a previou verion of a hp-fem code developed by C. Michler [8]. Figure 1 and diplay reult obtained by our hp-fem compared to thoe delivered by Michler oftware. We diplay the frequency pectra for two elected tet problem, and we oberve a perfect match between the olution ariing from the two different oftware package. In Fig. 3, we compare diperion curve againt Jun Ma oftware reult for a tet problem coniting of a tool equipped with a dipole ource and radiating in a fat homogenou formation. Figure 4, 5, and 6 illutrate the poibility of accurately reproducing diperion curve uing a limited number of frequencie (for the purpoe of thi paper, we have imply employed uniformly paced frequencie). Light-grey curve are the reult obtained from Jun Ma 1D emi-analytical code [11], while the black dot correpond to diperion data obtained by pot-proceing frequency-domain reult obtained with our new verion of the hp-fem oftware. To obtain the diperion curve uing our hp-fem oftware, we have employed only a limited number of frequencie. Specifically, we have employed 50, 5 and 10 frequencie for Figure 4, 5, and 6, repectively. From

3 Matuzyk et al. thee numerical reult, we conclude that when uing only a limited number of frequencie, reult till coincide with the analytical one. Furthermore, both the value of V p and V can be extracted from the reult, even when a low number of frequencie have been employed. CONCLUSIONS We have developed a new hp-fem method for imulation of acoutic meaurement baed on the ue of diperion curve. Thi method ha proven to be highly accurate and efficient, ince only a limited number of frequencie are needed to accurately reproduce diperion curve, which are ued for the aement of material propertie within the reervoir. The method can be readily extended to imulate borehole-eccentered tool and/or deviated well. ACKNOWLEDGEMENTS The work reported in thi paper wa funded by The Univerity of Texa at Autin Reearch Conortium on Formation Evaluation, jointly ponored by Anadarko, Aramco, Baker Hughe, BHP Billiton, BG, BP, Chevron, ConocoPhillip, ENI, ExxonMobil, Halliburton, He, Marathon, Mexican Intitute for Petroleum, Nexen, Petrobra, RWE, Schlumberger, StatoilHydro, TOTAL, and Weatherford. REFERENCES [1] Tang, X.-M. and Cheng, A., Quantitative borehole acoutic method, Handbook of geophyical exploration, Seimic Exploration, Vol. 4, ed. Helbig, K. and Treitel, S., Elevier, 004. [] Schmitt, D.P. and Bouchon, M., Full-wave acoutic logging: ynthetic microeimogram and frequency-wavenumber analyi, Geophyic, Vol. 50 (1985), no. 11, [3] Cheng, A.C.H. and Blanch, J.O., Numerical modeling of elatic wave propagation in a fluid-filled borehole, Communication in Computational Phyic, Vol. 3 (008), no. 1, [4] Chen, Y.-H., Chew, W.C. and Liu, Q.-H., A three-dimenional finite difference code for the modeling of onic logging tool, J. Acout. Soc. Am., Vol. 103 (1998), no., [5] Liu, Q-H., Schoen, E., Daube, F., Randall, C., and Lee, P., A three-dimenional finite difference imulation of onic logging, J. Acout. Soc. Am., Vol. 100 (1996), no. 1, [6] Liu, Q.H. and Sinha, B.K., A 3D cylindrical PML/FDTD method for elatic wave in fluid-filled preurized borehole in triaxially treed formation, Geophyic, Vol. 68 (003), no. 5, [7] Zheng, Y., Huang, X., and Tokoz, M.N., A finite element analyi of the effect of tool eccentricity on wave diperion propertie in borehole acoutic logging while drilling, Proceeding of the 74th Annual Meeting of the Society of Exploration Geophyicit, Denver, CO, 004. [8] Michler, Ch., Demkowicz, L., and Torre-Verdin, C., Numerical imulation of borehole acoutic logging in the frequency and time domain with hp-adaptive finite element, Computer Method in Applied Mechanic and Engineering, Vol. 198, Iue 1-6, Advance in Simulation-Baed Engineering Science - Honoring J. Tinley Oden, 1 May 009, Page , ISSN [9] Ihlenburg, F., Finite Element Analyi of Acoutic Scattering, Applied Mathematical Science Vol. 13, Springer, [10] Michler, C., Demkowicz, L., Kurtz, J., and Pardo, D., Improving the performance of Perfectly Matched Layer by mean of hp-adaptivity, Numerical Method for Partial Differential Equation vol. 3 (007), no. 4, [11] Ma, J., and Torre-Verdin, C., Radial 1D Simulation of Multipole Sonic Waveform in the Preence of a Centered, Non- Rigid Tool and Tranverely Iotropic Elatic Formation, The 8 th Reearch Conortium on Formation Evaluation, ed. C. Torre-Verdin, 008 3

4 Fig.. Comparion of the frequency pectrum for a dipole ource in a homogenou fat formation without a tool: reult obtained with Michler [8] (upper panel) and our new hp-fe code (lower panel). Fig. 3. Comparion of the diperion curve obtained for a = 50Hz: reult obtained with Jun Ma code [11] and our Fig. 1. Comparion of the frequency pectrum for a monopole ource in an open borehole for a layered formation: reult obtained with Michler [8] (upper panel) and our new hp-fe code (lower panel). Fig. 4. Comparion of the diperion curve obtained for a = 500Hz: reult obtained with Jun Ma code [11] and our

5 Matuzyk et al. Fig. 5. Comparion of the diperion curve obtained for a = 150Hz: reult obtained with Jun Ma code [11] and our Fig. 6. Comparion of the diperion curve obtained for a = 500Hz: reult obtained with Jun Ma code [11] and our 5