2D Finite Element Simulation of Sonic Measurements ...

2D FINITE-ELEMENT NUMERICAL SIMULATION OF SONIC MEASUREMENTS ACQUIRED IN THE PRESENCE OF A MANDREL TOOL Paweł J. Matuszyk1, Leszek Demkowicz1, David Pardo2, Jun Ma1, and Carlos Torres-Verdín1, 1 The University of Texas at Austin, 2Basque Center for Applied Mathematics, Bilbao, Spain

ABSTRACT We present a method for the simulation of sonic logging measurements using frequency-domain-based computations. A limited number of frequencies (typically below 50) are sufficient to accurately reproduce dispersion curves, which are used for the assessment of oil- and gas-bearing formations. We use an hp-FEM refinement strategy to perform highly-accurate frequency domain simulations. Verification results are displayed, both in terms of the frequency spectrum as well as in terms of dispersion curves. Numerical results illustrate the flexibility of the method, which can be employed to assess formations containing fractures and/or thin sand layers, and can be readily extended to the case of borehole-eccentered tools and/or deviated wells. INTRODUCTION Acoustic logging has been extensively used by oil companies to measure formation material properties such as porosity and mechanical rigidity. In order to understand and study the acquisition of sonic logging measurements, a plethora of numerical methods have been developed, including semi-analytical methods [1, 2, 3, 11], and timedomain finite differences [4, 5, 6]. Numerical simulations of sonic logging measurements have been almost exclusively performed in the time-domain (a few exceptions can be found in the literature, e.g, [7, 8]). Furthermore, frequency-domain based methods have lacked competitiveness with respect to time-domain methods, since a large number of frequencies need to be computed in order to obtain an accurate time response. Figures 1 and 2 display two typical spectra of sonic logging measurements, and illustrate the rapid variations that occur as we modify the frequency. These rapid variations in the amplitude of the pressure as a function of frequency demand the use of a large number of densely located frequencies (over 500) in order to perform an accurate inverse (fast) Fourier transform. We propose a new approach for performing frequency-domain based simulations of sonic logging measurements. The main idea of this paper is to obtain directly dispersion curves from frequency domain results, without ever obtaining a time-domain signal. After all, dispersion curves already contain the information about the slowness of the formation, which is employed by petro-physicists to obtain physical properties of the reservoir. Since dispersion curves are smooth with respect to variations in frequency (see Fig. 3), it is enough to calculate results only for a limited number of frequencies (below 50) to accurately reproduce dispersion curves. In addition, we believe that it is possible to further reduce the number of computed frequencies to a number below 15 when only Vp (high frequencies) and Vs (low frequencies) are needed. Furthermore, in the computer-aided simulations there is neither need to utilize a Ricker wavelet nor any other wavelet, because for each frequency, we only need information about the response of the sonic tool for a source with an arbitrary magnitude. Other advantages of using a frequencydomain-based approach include: (a) the added stability achieved when simulating the problem, since there is no need to impose small time-steps to assure stability of the problem, (b) the superior performance achieved by Perfectly Matched Layers (PML’s) in frequency domain computations (there is no need to worry about causality), and (c) the smaller computational complexity of the simulation problem associated to frequency-domain-based computations. On the other hand, the main challenges of frequency-domain-based simulations of sonic logging measurements include: (1) the multi-physics nature of the problem (coupled acoustics and elasticity), (2) the truncation of the computational domain, which involves the use of PML´s, (3) the need to properly treat different sources such as monopoles, dipoles, and quadruples, and (4) the so-called dispersion error associated to the simulation of highfrequency problems, which produces a delay of the traveling wave in the numerical solution. We note that the first three challenges are common to the use of time-domain-based simulations.

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METHOD In order to overcome the above challenges, we propose the use of a fully automatic hp-adaptive multi-physics finite element method (hp-FEM). This software incorporates a multi-physics automatic hp adaptive algorithm, which optimally adapts the mesh for each frequency, both with respect to element size h and polynomial order of approximation p, delivering exponential convergence rates in all simulation problems. In other words, this method delivers highly accurate solutions using a minimal number of degrees of freedom in comparison to other methods that employ other types of adaptation (h, p or r). The adaptive algorithm employs in each step a “two-grid strategy” (coarse and fine mesh) to estimate the error of the solution, which is utilized to make decisions about optimal mesh refinements. This hp-FEM software is utilized to accurately and efficiently simulate borehole acoustic measurements. The use of high-order methods (p>1) drastically reduces the dispersion error [9], which enables one to obtain accurate simulation results. In addition, the hp-FEM is ideally suited to solve boundary layers that necessarily arise from the use of PML’s [8, 10], which we employ to truncate the computational FE domain. In this work, we have developed a new version of the hp-FEM software enabling modeling of physical phenomena that is described by a set of coupled partial differential equations (PDE´s). In the case of sonic logging measurements, we have two different domains: (a) the acoustic domain, composed of the borehole fluid, and (b) the elastic domain, composed by the tool, formation, and possibly casing. FORMULATION Mathematically, we solve the following set of coupled PDE’s in the frequency domain (quantities p, u, σ are the Fourier transforms of the pressure, displacement, and stress tensor, respectively):

p 

2 p0 c 2f

￯ ￯ ￯ ��    s 2 u  0 ￯    I �� u    �u  �T u  ￯ ￯ n � p   s 2 n f � u f � ￯ ￯ ns �    pns where cf is the fluid speed sound, ω is the frequency, ρf and ρs are the fluid and solid densities, respectively, nf and ns are the outside normal vectors for the acoustic and elastic domains, respectively,

   s  V p2  2Vs2  and

   sVs2 are the so-called Lamé constants (Vp and Vs are the P- and S- wave velocities in the elastic solid, respectively). In the above set of equations, the first one describes the pressure phenomena in the acoustic domain, the second one is a momentum equation for the solid (linear elasticity), the third one is a constitutive equation, while the last two equations describe the coupling existing between both the acoustic and elastic domains. NUMERICAL RESULTS: The new code has been compared against several alternative numerical simulation methods such as a 1D semianalytical code developed by Jun Ma [11], and a previous version of a hp-FEM code developed by C. Michler [8]. Figures 1 and 2 display results obtained by our hp-FEM compared to those delivered by Michler´s software. We display the frequency spectra for two selected test problems, and we observe a perfect match between the solutions arising from the two different software packages. In Fig. 3, we compare dispersion curves against Jun Ma´s software results for a test problem consisting of a tool equipped with a dipole source and radiating in a fast homogenous formation. Figures 4, 5, and 6 illustrate the possibility of accurately reproducing dispersion curves using a limited number of frequencies (for the purpose of this paper, we have simply employed uniformly spaced frequencies). Light-grey curves are the results obtained from Jun Ma´s 1D semi-analytical code [11], while the black dots corresponds to dispersion data obtained by post-processing frequency-domain results obtained with our new version of the hp-FEM software. To obtain the dispersion curves using our hp-FEM software, we have employed only a limited number of frequencies. Specifically, we have employed 50, 25 and 10 frequencies for Figures 4, 5, and 6, respectively. From

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these numerical results, we conclude that when using only a limited number of frequencies, results still coincide with the analytical ones. Furthermore, both the values of Vp and Vs can be extracted from the results, even when a low number of frequencies have been employed. CONCLUSIONS We have developed a new hp-FEM method for simulation of acoustic measurements based on the use of dispersion curves. This method has proven to be highly accurate and efficient, since only a limited number of frequencies are needed to accurately reproduce dispersion curves, which are used for the assessment of material properties within the reservoir. The method can be readily extended to simulate borehole-eccentered tools and/or deviated wells. ACKNOWLEDGEMENTS The work reported in this paper was funded by The University of Texas at Austin’s Research Consortium on Formation Evaluation, jointly sponsored by Anadarko, Aramco, Baker Hughes, BHP Billiton, BG, BP, Chevron, ConocoPhillips, ENI, ExxonMobil, Halliburton, Hess, Marathon, Mexican Institute for Petroleum, Nexen, Petrobras, RWE, Schlumberger, StatoilHydro, TOTAL, and Weatherford. REFERENCES [1] Tang, X.-M. and Cheng, A., “Quantitative borehole acoustic methods”, Handbook of geophysical exploration, Seismic Exploration, Vol. 24, ed. Helbig, K. and Treitel, S., Elsevier, 2004. [2] Schmitt, D.P. and Bouchon, M., “Full-wave acoustic logging: synthetic microseismograms and frequency-wavenumber analysis”, Geophysics, Vol. 50 (1985), no. 11, 1756-1778. [3] Cheng, A.C.H. and Blanch, J.O., “Numerical modeling of elastic wave propagation in a fluid-filled borehole”, Communications in Computational Physics, Vol. 3 (2008), no. 1, 33-51. [4] Chen, Y.-H., Chew, W.C. and Liu, Q.-H., “A three-dimensional finite difference code for the modeling of sonic logging tools”, J. Acoust. Soc. Am., Vol. 103 (1998), no. 2, 702-712. [5] Liu, Q-H., Schoen, E., Daube, F., Randall, C., and Lee, P., “A three-dimensional finite difference simulation of sonic logging”, J. Acoust. Soc. Am., Vol. 100 (1996), no. 1, 72-79. [6] Liu, Q.H. and Sinha, B.K., “A 3D cylindrical PML/FDTD method for elastic waves in fluid-filled pressurized boreholes in triaxially stressed formations”, Geophysics, Vol. 68 (2003), no. 5, 1731-1743. [7] Zheng, Y., Huang, X., and Toksoz, M.N., “A finite element analysis of the effects of tool eccentricity on wave dispersion properties in borehole acoustic logging while drilling”, Proceedings of the 74th Annual Meeting of the Society of Exploration Geophysicists, Denver, CO, 2004. [8] Michler, Ch., Demkowicz, L., and Torres-Verdin, C., “Numerical simulation of borehole acoustic logging in the frequency and time domains with hp-adaptive finite elements”, Computer Methods in Applied Mechanics and Engineering, Vol. 198, Issues 21-26, Advances in Simulation-Based Engineering Sciences - Honoring J. Tinsley Oden, 1 May 2009, Pages 18211838, ISSN 0045-7825. [9] Ihlenburg, F., “Finite Element Analysis of Acoustic Scattering”, Applied Mathematical Sciences Vol. 132, Springer, 1998. [10] Michler, C., Demkowicz, L., Kurtz, J., and Pardo, D., “Improving the performance of Perfectly Matched Layers by means of hp-adaptivity”, Numerical Methods for Partial Differential Equations vol. 23 (2007), no. 4, 832-858. [11] Ma, J., and Torres-Verdin, C., “Radial 1D Simulation of Multipole Sonic Waveforms in the Presence of a Centered, NonRigid Tool and Transversely Isotropic Elastic Formations”, The 8th Research Consortium on Formation Evaluation, ed. C. Torres-Verdin, 2008

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Fig. 2. Comparison of the frequency spectrum for a dipole source in a homogenous fast formation without a tool: results obtained with Michler’s [8] (upper panel) and our new hp-FE code (lower panel).

Fig. 3. Comparison of the dispersion curves obtained for a problem with a layered formation and a frequency spacing df = 50Hz: results obtained with Jun Ma´s code [11] and our new hp-FE code (PJM). Fig. 1. Comparison of the frequency spectrum for a monopole source in an open borehole for a layered formation: results obtained with Michler’s [8] (upper panel) and our new hp-FE code (lower panel).

Fig. 4. Comparison of the dispersion curves obtained for a problem with a layered formation and a frequency spacing df = 500Hz: results obtained with Jun Ma´s code [11] and our new hp-FE code (PJM).

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Fig. 5. Comparison of the dispersion curves obtained for a problem with a layered formation and a frequency spacing df = 1250Hz: results obtained with Jun Ma´s code [11] and our new hp-FE code (PJM).

Fig. 6. Comparison of the dispersion curves obtained for a problem with a layered formation and a frequency spacing df = 2500Hz: results obtained with Jun Ma´s code [11] and our new hp-FE code (PJM).

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