We Simultaneous Joint Inversion of Electromagnetic and Seismic Full-waveform Data - A Sensitivity Analysis to Biot Parameter
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1 We Simultaneous Joint Inversion of Electromagnetic and Seismic Full-waveform Data - A Sensitivity Analysis to Biot Parameter J. Giraud* (WesternGeco Geosolutions), M. De Stefano (WesternGeco Geosolutions) & F. Miotti (WesternGeco Geosolutions) SUMMARY We present and compare different methods for computing Gassmann s fluid substitution model in the framework of the petrophysical simultaneous joint inversion of electromagnetic and seismic data. The algorithm we used computes elastic parameters using Gassmann s model, while electrical resistivities are calculated using Archie s law. In Gassmann s model, the formulation of the Biot coefficient is critical to compute elastic properties of the saturated medium. Concerning the formulation of the Biot coefficient, along with testing the simultaneous joint inversion using expressions proposed in the models of Krief, Nur, Pride and Lee we also derived and tested a modified version of Krief s model, which is inspired by that of Pride and Lee. As a result, for comparable misfits, Nur s critical porosity model permits us to retrieve well the geometries of the medium, while Krief s model allows us to retrieve sharper boundaries. Also, our study permits us to determine the influence of the formulation of the Biot parameter on inverted electrical resistivity distribution. In comparison to other models, the Pride and Lee model delineates quite well geometries of the synthetic model while being less efficient to retrieve electrical resistivity distribution. The modified Krief s model provided slightly improved misfit with respect to Krief s model.
2 Introduction Lately, the integration of different methods has taken a growing importance in oil and gas E&P. Most recently, the joint inversion of multiple data sets was introduced by De Stefano et al. (2011) for the structural constraint. Subsequently, Gao et al. (2012) and Abubakar et al. (2012) presented a petrophysical simultaneous joint inversion approach, upon which this paper is focused. We present an extension of the existing work on the petrophysical side by testing additional petrophysical formulas in the framework of the simultaneous joint inversion (SJI). When considering petrophysical approaches, using an appropriate petrophysical model in real-world reservoir characterization is crucial. As is illustrated in this work, even slightly wrong petrophysical models can lead to poor results. Starting from the latter two authors work we extended it following the study of the efficiency and accuracy of Krief et al. (1990), Nur (1992), Pride et al. (2004), and Lee (2005) models of Biot coefficient formulations performed by Zhang et al. (2009). Deriving the models of Pride and Krief, we also introduce a modified version of Krief s model. Inversion algorithm The inversion algorithm we used aims at the minimization of a multiplicative joint cost function:, (1) where and are the weighted objective functions for seismic and electromagnetic (EM) data, respectively. Superscript denotes electromagnetic while superscript denotes seismic. The factor is a so-called multiplicative regularization function. Seismic data are frequency weighted. is the model at the -th iteration. For each cell of the 2D medium, it consists in the porosity and water saturation. The cost function is minimized thanks to a Gauss-Newton minimization approach. The oil saturation ( ) is simply computed as. The seismic displacement field is obtained by solving the elastic equation at each frequency while the electric field is calculated by solving Maxwell s equation. In both cases calculations are performed using a frequency-domain finite-difference approach. Computing geophysical properties, linking EM and seismic data Electrical conductivities are obtained by computing Archie s law (Archie 1942):, (2) where is the electrical conductivity of the medium and is that of water, is the cementation exponent, is a parameter related to tortuosity, and is a constant usually depending on the medium. Seismic velocities are computed using Gassmann s fluid substitution model (Gassmann 1951). For the purpose of our study, we reformulate it as follows: (3). (5) Porosity (ϕ), and saturations ( refers to the saturations of the -th fluide phase) are part of the model, in which is the bulk modulus, µ is the shear modulus, and is the mass density. Subscripts,, and refer to matrix, saturated medium, and fluid, respectively. M is the grain bulk modulus (4)
3 while is the total number of fluid phases present in the porous space. For a given model, and are the formulation of a Biot coefficient applied to the computation of bulk and shear modulus, respectively. Except for the Pride and Lee and modified Krief s models,. In our study, we used the four following formulations of the Biot coefficient: Nur s model, Krief s model, the model of Pride and Lee, and the modified Krief s model. The model of Nur (1992), for which, is given as: (6) Here, is the critical porosity. The model of Krief (Krief et al. 1990), in which, is defined as:, where, is an empirical coefficient. This value varies with the medium and can be set in accordance with the lithology. The model of Pride and Lee is formulated as follows (Lee (2005), Pride et al. (2004)):, (8) where and are empirical multiplicative factors. is set to 1.5 by Pride et al. (2004) (original formulation of Pride s model), while Lee (2005) redefined it as a function of (this led to the socalled model of Pride and Lee). Last, Krief s modified model is the same as Krief s original model when addressing the computation of. However, it differs for the expression of, which we introduce as:, (9) where is an empirical value sought to take into account the effect of the presence of fluids. Beside plots of the inverted properties over the inversion area, interpretation of inversion results is made by the use of the basic statistical quantities such as the misfit between inverted and true properties and the Bravais-Pearson linear correlation coefficient. Comparing inversion results obtained using different Biot parameter formulations After studying the influence of Biot parameter formulation on elastic moduli of the medium, we studied the ability of each particular formulation to retrieve the synthetic model through SJI. For each particular formulation, we generated a synthetic data set, which we inverted using different implementations of the Biot parameter. The most relevant properties of our study are water saturation and porosity. Because they are the two properties that are obtained through the SJI, we chose to show the latter two inverted properties only. (7)
4 Example: inverting data generated using modified Krief s model Figure 1 displays true and inverted properties. Figure 1 True (on top) and inverted porosity and oil saturation ((a) to (d)). Data are generated using the modified Krief s model while the other parameters are kept constant. Inverted models: (a) using Nur s model during inversion, (b) using Krief s model during inversion, (c) using the model of Pride and Lee during inversion, (d) using modified Krief s model. The depth of the seawater is 1 km. Both reservoirs have a porosity of 0.25 and a water saturation of 0.2. The two inhomogeneities below the reservoirs have a porosity of 0.1 and 0.15, respectively. Electrical resistivities (equation (2)) are calculated using the following Archie's parameters: =1, =0.4, =2.4. Nur s model critical porosity (equation (6)) is set to 0.40, Krief s empirical exponent is set to 2.75 (equation (7)) while is set to 1.5 into Pride s model (equation (8)). For the use of the modified Krief s model (equation (9)), a value of 2.6 and 0.25 for and, respectively, was used. Seismic data consist in vertical and horizontal displacement fields. Frequencies for which data are simulated are 1 Hz, 2 Hz, and 3 Hz. EM data are simulated for x-oriented dipoles. Frequencies for which EM data are simulated are Hz, 0.25 Hz, and 1 Hz. In Figure 1, receivers and transmitters are separated by 500 m. There are 41 transmitter positions and 40 receivers. From Figure 1, it is worth pointing out that, when the modified Krief model is used, even though it is close to Krief s, the results obtained with the latter are closer to those obtained by inverting thanks to Nur s model. One of the main interesting features obtained from the testing is that, for a comparable final cost function, results obtained through the use of the modified Krief s model show a better agreement with the synthetic model than for any other model of the Biot parameter. The table below presents a summary of the comparison of the Biot parameter formulations.
5 Table 1 Comparison of the efficiency of the formulations of Biot parameter Model used to generate the data Best model for inverting second best model for inverting second worst model for inverting worst model for inverting Nur Nur modified Krief Krief Pride and Lee Krief modified Krief Krief Nur Pride and Lee Pride and Lee Pride and Lee modified Krief Krief Nur modified Krief modified Krief Krief Nur Pride and Lee Conclusions Generally speaking, structures of the inverted model obtained by using the modified Krief s model seem to be better retrieved than when using other formulations. Misfits and correlations are practically always better when the modified Krief s model is employed. On the other hand, Nur s model seems quite appropriate to most inversions we ran in the framework of the testing of the efficiency of the different formulations we presented in this work. A similar conclusion can be drawn for Krief s model. Contrarily, the model of Pride and Lee provides inversion results that were not as good. Thus, this study seems to indicate that, when addressing real data, Krief s and modified Krief s models may be the most robust. In our work, we considered a limited number of the possible values of the empirical parameters. Additional tests involving a stochastic approach, varying the empirical parameters of the Biot parameter formulations, may be required to make this study more exhaustive. Also, additional petrophysical relationships could be tested in further work. Acknowledgements The authors thank Schlumberger-Doll Research for providing the SJI code as well as the synthetic model we used for our testing and for their support through regular meetings throughout the project. References Abubakar, A., Gao, G., Habashy, T. M., and Liu, J. [2012] Joint inversion approaches for geophysical electromagnetic and elastic full-waveform data. Inverse Problems, 28. Archie, G. E. [1942] The electrical resistivity log as an aid in determining some reservoir characteristics. Petroleum transactions of AIME, 168, De Stefano, M., Golfré Andreasi, F., Re, S., Virgilio, M., and Snyder, F. F. [2011] Multiple-domain, simultaneous joint inversion of geophysical data with application to subsalt imaging. Geophysics, 76 (3), R69-R80. Gao, G., Abubakar, A., and Habashy, T. M. [2012] Joint Inversion of electromagnetic and fullwaveform seismic data. Geophysics, 77 (3), wa3-wa18. Gassmann, F. [1951] Uber die Elastizitat poroser Medien: Vierteljahrsschrift der Naturforschenden. Gesellschaft, 96, Krief, M., Garat, J.,Stellingwerff, J., and Ventre, J. [1990] A petrophysical interpretation using the velocities of P and S waves (full-waveform sonic). The Log Analyst, 31, Lee, M. W. [2005] Proposed moduli of dry rock and their application to predicting elastic velocities of sandstones. U. S. Geological Survey, Scientific Investigations Report, Nur, A. [1992] Critical porosity and the seismic velocities in rocks. Eos, Transactions, American Geophysical Union, 73. Pride, S. R., Berryman, J. G., and Harris, J. M. [2004] Seismic attenuation due to wave-induced flow. Journal of Geophysical Research, 109, B B Zhang, J., Li, H., Liu, H., and Cui, X. [2009] Accuracy of Krief, Nur and Pride models in the study of rock physics. 79 th Annual International Meeting, SEG, Expanded Abstracts,
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