Joint inversion of borehole electromagnetic and sonic measurements G. Gao, A. Abubakar, T. M. Habashy, Schlumberger-Doll Research SUMMARY New-generation electromagnetic (EM) and sonic logging tools are capable of measuring multi-frequency multi-component data at multiple depths of investigation. This allows us to image the formation around the borehole by using an inversion method. A method is developed for directly obtaining porosity and fluid saturation distributions by simultaneously inverting borehole EM and sonic measurements. This joint inversion reduces the nonuniqueness of determining porosity and fluid saturation distributions, which can not be achieved with either sonic or EM inversion only. We show that the inversion algorithm can accurately obtain porosity and water saturation distributions around the borehole including the zones invaded by the mud filtrate, oil-water contacts, and as well as the dipping structures. INTRODUCTION Accurate determination of formation petrophysical parameters, such as porosity and fluid saturation distributions around the borehole, is the main goal of petroleum formation evaluation (Schlumberger, 1991). Particularly crucial are those in zones that are not affected by drilling-mud-filtrate invasion around the borehole. In addition, monitoring the fluid movement is essential during the water flooding process and petroleum production. Porosity is routinely determined by borehole sonic measurements, and fluid saturations are routinely calculated from borehole resistivity measurements employing an empirical saturation equation, such as Archie s equation, Waxman- Smits equation, and others (Schlumberger, 1991). Porosity and fluid saturations determined in this way are usually a volumetric average around the borehole. Radial models are sometimes used to account for the mud-filtrate invasion to derive the formation resistivity beyond the invasion zone. Nevertheless, those profiles are not a true representation of the porosity and fluid saturation around the borehole, radially, and azimuthally because they are essentially distributed around the borehole inhomogeneously. As a result, it is preferable to have porosity and fluid saturation distributions rather than a volumetric average. New-generation logging tools, such as the triaxial induction tool used in Leveridge (0) and sonic tool described in Pistre et al. (005), are capable of making 3D measurements at multiple depths of investigation. The triaxial induction tool collects multi-frequency EM measurements using multiple triaxial arrays, each containing three collocated coils, which are used for more accurate determination of the water saturation because the resistivity anisotropy can be quantified (Leveridge, 0). The sonic tool makes both monopole and dipole measurements in a wide frequency range for axial, azimuthal, and radial information near the wellbore. The sonic tool can also see past the altered zone and provides measurements for the unaltered zone (Pistre et al., 005). These added measurements make it possible to derive porosity and fluid saturation images around the borehole, which could result in enhanced porosity and hydrocarbon saturation estimates, thus leading to an improved reservoir characterization. On the other hand, in the routine formation evaluation, porosity and fluid saturations are determined separately from different measurements. However, those measurements may be sensitive to both porosity and fluid saturations. As a result, to improve the determination of porosity and fluid saturation, it is better to determine them simultaneously from multi-physics measurements. In this paper, we will present a method that can determine porosity and fluid saturations distributions around the borehole simultaneously from borehole EM and sonic data. It has been shown that, by making use of petrophysical links, the joint inversion of EM and seismic measurements cannot only reduce the nonuniqueness for the determination of in-situ formation porosity and fluid saturation distributions, but it also improves the inversion results (Hoversten et al., 00; Wang et al., 007; Gao et al., 0, 01). As a result, the joint petrophysical approach is becoming a very useful tool for reservoir characterization and monitoring applications. In our inversion algorithm, for illustration purposes, we use Archie s equation (Archie, 19) to link conductivity to porosity and fluid saturations, and the fluid substitution equations (Gassmann, 1951) coupled with the critical porosity model (Nur, 199) to link the seismic velocities and mass density to porosity and saturations. We previously did an analysis on how the accuracy of these petrophysical links affect the inversion results(gao et al., 01). In this paper, we assume that those petrophysical relationships are known. We present a numerical example to show that the proposed inversion approach can help efficiently and accurately determine porosity and fluid saturation distributions around a borehole including dipping layers and oil-water contacts. THEORY Petrophysical Relationships Archie s equation Archie (19) discovered that the formation conductivity (σ) can be expressed as a function of the porosity (φ) and the water saturation (S w ) according to the relationship: σ = 1 a σ wφ m S n w, (1) where a is a tortuosity factor, m is the porosity/cementation exponent, n is the saturation exponent, and σ w is the conductivity of the formation saline water. Archie s equation is valid for clean sand formations. Since the development of Archie s equation, other variants have been introduced to account for the conduction of the clay content in rocks, such as Waxman- Smits equation (Waxman and Smits, 19). Archie s equation 01 SEG SEG Las Vegas 01 Annual Meeting Page 1
Joint inversion of borehole EM and sonic measurements and its variants have served as important tools to calculate the hydrocarbon saturation from EM measurements. Gassmann s equations On the other hand, the fluid substitution model (Gassmann, 1951) links seismic velocities to reservoir parameters, such as porosity (φ), water saturation (S w ), and oil saturation (S o ) or gas saturation (S g ). In fluid-saturated rocks, the compressional wave (P-wave) velocity V P and the shear-wave velocity V S are expressed as follows: K sat + 3 V P = µ sat, () ρ sat µsat V S =, (3) ρ sat where K sat = (1 β)k ma + β M, () µ sat = (1 β)µ ma, (5) ( β φ M = + φ ) 1, () K ma K f ( ) 1 S w S o S g K f = C w +C o +C g, (7) K w K o K g ρ sat = (1 φ)ρ ma + φ ( S w ρ w + S o ρ o + S g ρ g ). () In equations, β is the Biot coefficient, which in general is a function of porosity. In this study, we chose the critical porosity model of Nur (199) for β, { φ/φc, 0 φ φ β = c (9) 1, φ > φ c where φ c is the critical porosity above which solids become suspensions. In the above equations, K sat, µ sat, and ρ sat are the bulk modulus, the shear modulus, and the bulk density of the fluid-saturated rock, K f is the bulk modulus of the pore fluid; K ma, µ ma, and ρ ma are the bulk modulus, shear modulus, and the density of the matrix (solid or grain), and K w, K o, and K g are the bulk modulus of water, oil, and gas, respectively. ρ w, ρ o, and ρ g are the density of water, oil, and gas, respectively. C w, C o, and C g are correction terms for water, oil, and gas, respectively. Note that the gas correction was found to be necessary by Hoversten et al. (00) in the existence of gas, while C w and C o are usually set to unity. Inversion Methodology In the example presented in this paper, as forward solvers, we employ a.5d frequency-domain finite-difference method for EM simulation (Abubakar et al., 00) and a D frequencydomain finite-difference approach for elastic wave simulation (Abubakar et al., 0). Assume that m φ and m S w are the unknown vectors representing the porosity and water saturation distributions, respectively; then, the total unknown vector is m = [ m φ m S w] T, where T denotes the matrix transpose. The cost function for the single-physics petrophysical inversion is given by Φ(m) = Φ d (m) + λ Φ m (m), () which is, for the Gauss-Newton method, equivalent to the multiplicative cost function (as introduced by van den Berg et al., 1999) when the regularization parameter λ at each iteration n is set to be λ n = Φ d (m n ) Φ m (m n ). (11) The normalized data misfit Φ d is given by Φ d (m) = 1 N F NS NR [ (η k ) i=1 j=1 wd;i, j,k di, j,k s i, j,k (m) ] NS NR k=1 i=1 j=1 wd;i, j,k d i, j,k = 1 W d [d s(m)], (1) where N F, N S, and N R are the number of frequencies, sources, and receivers, respectively. The vector d is the measured EM or sonic data and s is the simulated response vector for a given model parameter m. The matrix w d;i, j,k is the data weighting matrix whose diagonal elements are the inverse of the estimates of the standard deviation of the measurement noise. The frequency weighting factor η k is used for the sonic inversion as a way to prevent the high-frequency components from dominating the inversion process. The regularization cost function is either the l -norm or the weighted l -norm as described in Abubakar et al. (00). The cost function for the joint petrophysical inversion is written as follows: Φ(m) = Φ SM d (m) + Φ EM d (m) + λφ m (m), (13) where Φ SM d is the sonic data misfit and Φ EM d is the EM data misfit. They are defined in equation 1. For both the singlephysics inversion and the simultaneous joint inversion, we employ the Gauss-Newton minimization framework described in Habashy and Abubakar (00). The detailed algroithm for the petrophysical inversion is described in Gao et al. (01). In our implementation, the forward response, the Jacobian construction, and the conjugate gradient least square (CGLS) iterative process which is used to solved the linearized system, are parallelized using the message passing interface (MPI) library and their memory usage are distributed nearly linearly among all processors, which is crucial for the application to an extremely large-scale problem. INVERSION EXAMPLE To demonstrate the proposed joint inversion approach, we show an inversion example using measurements from the triaxial induction tool used in Leveridge (0) and the sonic tool described in Pistre et al. (005). The purpose is to use the inversion approach to obtain the porosity and fluid saturation distributions around a borehole. Model setup Figure 1 shows porosity and water saturation distributions of the model. The model has layers with different values of porosity and water saturation. Especially, for the fourth layer, although the porosity remains the same, the bottom portion 01 SEG SEG Las Vegas 01 Annual Meeting Page
Joint inversion of borehole EM and sonic measurements 0.3 1 0. 0.00 3. 39.1 5000 1 1 0 0. 0.1 φ 0 1 1 0 0. 0. 0. 0 S w 1 1 5.1 15.5.00.31 3.9.51 1.5 0 R(Ω m) 1.00 (a) Resistivity 1 1 500 000 0 V p (m/s) 3500 (b) P-wave velocity (a) Porosity (b) Water saturation 3500 00 Figure 1: True petrophysical models. 500 below 11 m is saturated with water while the upper portion above 11 m is saturated with oil (0%). Because of the limitation of the depth of investigation for borehole logging tools, the model extension away from the borehole is 3 m. Figure shows the resisitivity, P-wave velocity, S-wave velocity, and density structures calculated from the porosity and water saturation distributions using the petrophysical relationships. For the petrophysical transforms, we use the following Archie s parameters: a = 1, m = 1., n =, and σ w =.0 S/m. In the rock physics model, the following parameters are used: K ma = 37 GPa, µ ma = GPa, K w =.5 GPa, K o = 0.75 GPa, ρ ma = 50 kg/m 3, ρ w = 50 kg/m 3, ρ o = 750 kg/m 3, and φ c = 0.. All fluid correction terms are set to 1. 1 1 3000 500 0 V s (m/s) 000 (c) S-wave velocity 1 1 0 (d) Mass density 00 300 00 0 000 ρ(kg/m 3 ) Survey setup For the EM survey, we use the tri-axial induction tool configuration. Basically it has three tri-axial magnetic transmitters and three tri-axial magnetic receivers for each transmitter-receiver pair. As a result, it can measure all nine-components of the magnetic field tensor in such a way that dipping and azimuthal angles, as well as electrical anisotropy can be determined to improve the formation evaluation. In our study, we use six transmitter-receiver spacings: 13.5 in, 19 in, in, 35 in, 9 in, and 70 in. The magnetic components of the magnetic field tensor used are H xx, H yy, H zz, H xz, and H zx. The operating frequencies of the transmitter are 15 khz and 30 khz. The logging interval is 0.3 m. For the acoustic survey, we use the sonic tool configurations shown in Figure 3. The tool can make monopole measurements (marked in blue) and dipole measurements (marked in red). The two dipole transmitters are called 0 degree and 90 degrees, which generate a dipole mode aligned to the tool refrence and 90 degrees to it (clockwise looking downward by convention), respectively. The tool features 13 axial stations Figure : True geophysical models. separated by in between them for a total aperture of ft for the receiver array. Eight azimuthal receivers are located every 5 degrees around the tool for each of the 13 stations providing a total of sensors for the whole receiver array. In our simulation, we did not consider the azimuthal differences for the receiver, which means for each station only one receiver is used, instead of eight. We process the acoustic data using a frequency-domain approach. The operating frequencies used are 0.5,.5, 5.0 and.0 khz. The logging interval is 0.3 m. For EM and sonic modeling, we use the same regular grid configuration: the grid size is 0.05 m in both the x- and z- directions. We added % Gaussian random noise to both the EM and sonic data. As initial models, we use a homogeneous model with 0.1 for the porosity and 0. for the water saturation. 01 SEG SEG Las Vegas 01 Annual Meeting Page 3
Joint inversion of borehole EM and sonic measurements Figure 3: Geometry of the sonic tool used in the example (Pistre et al., 005). 1 1 0 φ 0 0.3 0. 0.1 1 1 0 1 S w 0 0. 0. 0. 0. 1 1 0 (a) Resistivity 0.00 3. 39.1 5.1 15.5.00.31 3.9.51 1.5 R(Ω m) 1.00 3500 3000 1 1 5000 500 000 0 V p (m/s) 3500 (b) P-wave velocity 00 500 00 300 (a) Porosity (b) Water saturation 1 500 1 00 Figure : Inverted petrophysical model from joint inversion. 1 1 0 Joint inversion results Figure shows the inverted porosity and water saturation distributions using the joint inversion methodology, and Figure 5 shows the corresponding resistivity, P-wave velocity, S-wave velocity, and mass density distributions. After inversion, the sonic data misfit is reduced to 1.59%, and the EM data misfit is reduced to.1%, which means the joint inversion fits both the EM data and sonic data very well at the same time. Figure clearly shows that the joint inversion reconstructed both porosity and water saturation distributions very well, including the dipping feature and the oil-water contacts, which is further confirmed by the geophysical models shown in Figure 5. Moreover, we can observe the decrease of resolution away from the borehole due to the decrease of sensitivity with increasing distance from the borehole. Inversion results demonstrate that the proposed method can provide accurate estimates of porosity and fluid saturation distributions around the borehole, and inversion results can be significantly improved if both EM and sonic measurements are used simultaneously. CONCLUSIONS We developed a joint inversion approach to directly invert for porosity and saturation distributions around the borehole from 0 (c) S-wave velocity V s (m/s) 000 0 (d) Mass density 000 ρ(kg/m 3 ) Figure 5: Calculated geophysical model from the inverted petrophysical models for joint inversion. new-generation borehole electromagnetic and sonic measurements. Measurements from the tri-axial induction tool and the sonic tool described in Pistre et al. (005) are considered in the example. We showed that the inversion approach can accurately estimate the porosity and fluid saturation distributions around the borehole and identify oil-water contacts. This is very promising for improved formation evaluation. ACKNOWLEDGMENTS The authors thank V. Druskin from Schlumberger-Doll Research for providing the.5d EM forward modeling code and J. Liu from Schlumberger for his contributions on the D elastic inversion code. 01 SEG SEG Las Vegas 01 Annual Meeting Page
EDITED REFERENCES Note: This reference list is a copy-edited version of the reference list submitted by the author. Reference lists for the 01 SEG Technical Program Expanded Abstracts have been copy edited so t hat references provided with the online metadata for each paper will achieve a high degree of linking to cited sources that appear on the Web. REFERENCES Abubakar, A., T. M. Habashy, V. Druskin, D. Alumbaugh, and L. Knizhnerman, 00,.5D forward and inversion modeling for interpreting low -frequency electromagnetic measurements: Geophysics, 73, no., F5 F177. Abubakar, A., T. M. Habashy, Y. Lin, and J. Liu, 0, D elastic frequency -domain full-waveform seismic inversion using the varying PML approach: 7nd Conference and Exhibition, EAGE, Extended Abstracts. Archie, G. E., 19, The electrical resistivity log as an aid in determining some reservoir characteristics: Petroleum Transactions of AIME, 1, 5. Gao, G., A. Abubakar, and T. M. Habashy, 0, Joint inversion of cross -well electromagnetic and seismic data for reservoir petrophysical parameters: Annual Technical Conference and Exhibition, SPE, 135307. Gao, G., A. Abubakar, and T. M. Habashy, 01, Joint petrophysical inversion of electromagne tic and full-waveform seismic data: Geophysics, 77, no., D53. Gassmann, F., 1951, Uber die elastizit at poroser medien: Vierteljahrsschrift der Naturforschenden Gesellschaft, 9, 1 3. Habashy, T. M., and A. Abubakar, 00, A general framework for cons traint minimization for the inversion of electromagnetic measurements: Progress in Electromagnetic Research, PIER, 5 31. Hoversten, G. M., F. Cassassuce, E. Gasperikova, G. A. Newman, J. Chen, Y. Rubin, Z. Hou, and D. Vasco, 00, Direct reservoir pa rameter estimation using joint inversion of marine seismic AVA and CSEM data: Geophysics, 71, no. 3, C1 C13. Leveridge, R. M., 0, New resistivity-logging tool helps resolve problems of anisotropy, shoulder -bed effects: Journal of Petroleum Technology, 0 3. Nur, A., 199, Critical porosity and the seismic velocities in rocks: Eos, Transactions, American Geophysical Union, 73, 3. Pistre, V., T. Kinoshita, T. Endo, K. Schilling, J. Pabon, B. Sinha, T. Plona, T. Ikegami, and D. Johnson, 005, A modula r wireline sonic tool for measurements of 3D (azimuthal, radial, and axial) formation acoustic properties: th Annual Logging Symposium, Society of Petrophysicists and Well Log Analysts, Expanded Abstracts, Paper P. Schlumberger, 1991, Log interpretation principles/applications: Schlumberger. Van den Berg, P. M., A. L. Broekhoven, and A. Abubakar, 1999, Extended contrast source inversion: Inverse Problems, 15, 135 13. Wang, G. L., C. Torres-Verdin, J. Ma, and T. B. Odumosu, 007, Combined inversion of b orehole resistivity and sonic measurements to estimate water saturation, porosity, and dry-rock elastic moduli 01 SEG SEG Las Vegas 01 Annual Meeting Page 5
in the presence of invasion: th Annual Logging Symposium, Society of Petrophysicists and Well Log Analysts, Expanded Abstract, Paper LLL. Waxman, M. H., and M. J. L. Smits, 19, Electrical conductivities in oil -bearing shaly sands: SPE Journal,, 7 1. 01 SEG SEG Las Vegas 01 Annual Meeting Page