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1 SPE Joint Raial Inversion of Resistivity an Sonic Logs to Estimate In-Situ Petrophysical an Elastic Properties of Formations Robert K. Mallan, SPE, Jun Ma, Carlos Torres-Verín, SPE, an Gong Li Wang, SPE, The University of Texas at Austin Copyright 2009, Society of Petroleum Engineers This paper was prepare for presentation at the 2009 SPE Annual Technical Conference an Exhibition hel in New Orleans, Louisiana, USA, 4 7 October This paper was selecte for presentation by an SPE program committee following review of information containe in an abstract submitte by the author(s). Contents of the paper have not been reviewe by the Society of Petroleum Engineers an are subject to correction by the author(s). The material oes not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Electronic reprouction, istribution, or storage of any part of this paper without the written consent of the Society of Petroleum Engineers is prohibite. Permission to reprouce in print is restricte to an abstract of not more than 300 wors; illustrations may not be copie. The abstract must contain conspicuous acknowlegment of SPE copyright. Abstract Presence of near-wellbore amage, resulting from rilling an mu-filtrate invasion, can substantially affect sonic an resistivity borehole logging measurements. Therefore, unbiase log interpretation must account for presence of invasion in orer to procure accurate estimates of formation properties. Our objective is to estimate the formation s ry bulk an shear mouli, porosity, an water saturation from the joint inversion of borehole array-inuction resistivity an sonic measurements. We assume a raial one-imensional (1D) moel in the inversion, with the formation moel escribe by a raial variation of water an hyrocarbon saturations representative of mu-filtrate invasion. The inversion is guie by the ata misfit of both array-inuction apparent resistivities an sonic-log flexural an Stoneley wave velocity-frequency ispersion curves. Raial istributions of fluis are converte to istributions of resistivity, ensity, an bulk moulus, which are input to the simulations of apparent resistivity an sonic logs. We make use of flui-substitution moels to relate bulk ensity, ry bulk moulus, an ry shear moulus to porosity an flui saturation. Apparent resistivities are simulate base on a commercially available array-inuction logging tool. Sonic measurements are analyze in the frequency omain via flexural an Stoneley wave moe ispersions, which are calculate irectly in the frequency omain. Synthetic cases consier water-base mu filtrate invaing a hyrocarbon-bearing san an oil-base mu filtrate invaing a water-bearing san. Porosities an elastic properties consistent with a soft formation are consiere in the moels. Sensitivity analysis inicates that sonic flexural an Stoneley moe ispersions naturally complement apparent resistivity measurements in the presence of mu-filtrate invasion. Inversions of synthetic cases prouce reliable estimates of ry-rock bulk an shear mouli, porosity, an initial water saturation. Furthermore, these cases show that combining resistivity an sonic measurements reuces ambiguity in the inversion. Introuction Presence of near-well amage as a result of mu-filtrate invasion can substantially affect sonic an resistivity borehole logging measurements. Because of the strong effect of mu-filtrate invasion on resistivity an sonic logging measurements, presence of invasion must be consiere in the interpretation, or inversion, of these ata to obtain reliable estimates of elastic an petrophysical formation properties. Consequently, inversion must estimate the variation of the properties extening raially away from the borehole. Inepenent inversion of resistivity an sonic ata for this type of moel can be non-unique an thus require the enforcement of constraints, or a priori information. However, because these two measurements epen on ifferent physics, an they are interconnecte by the relationship between ensity an water saturation, their combination in a joint inversion scheme may suppress the ambiguity in the estimates of the formation properties. Furthermore, the joint inversion may suppress the nee for artificial constraints because the complementing measurements may act as a self constraint. Inversions of sonic log measurements have been propose for 1D raial profiling of shear mouli from flexural an Stoneley ispersions (Sinha et al., 2006) an for raial profiling of compressional an shear wave velocities an ensity from full waveform (Chi et al., 2004). Recent work has been performe on the combine inversion of borehole resistivity an sonic measurements to estimate the 1D raial profile of elastic an petrophysical properties of a formation having a piston-like invasion front (Wang et al., 2007), where sonic inversion was performe on the full waveform, an combine inversion use the solution from the separate inversion of resistivity measurements in the subsequent inversion of sonic measurements.

2 2 SPE We apply the Levenberg-Marquart nonlinear minimization metho in the joint inversion of numerically simulate flexural an Stoneley ispersions an array-inuction apparent resistivity measurements for the ry bulk an shear mouli, porosity, an the 1D raial profile of water saturation. Metho A raial 1D moel is assume that escribes the variation of formation properties consistent with mu-filtrate invasion into a san formation. We examine cases escribing water-base mu-filtrate (WBM) invaing a gas-bearing san an oil-base mufiltrate (OBM) invaing a water-bearing san. Furthermore, we assume elastic properties inicative of a slow formation. The inversion scheme employe is the Levenberg-Marguart metho. Unknown parameters are the raial istribution of water saturation, porosity, an ry-rock bulk an shear mouli. The 1D moel is parameterize into seven raial layers for both simulation an inversion. Parameterize raial layers are uniformly istribute an exten outwar to a istance of 1 m. Data riving the inversion are array-inuction raw conuctivities an sonic velocity-frequency ispersions. Raw, borehole correcte conuctivities are simulate base on the Schlumberger AIT-H inuction logging tool. All sixteen channels, comprising real an imaginary conuctivities, are use in the inversion. Flexural an Stoneley ispersions are simulate using an analytic ispersion coe. The parameters φ (porosity) an S w (water saturation) are relate to electrical conuctivity through Archie s equation, namely, 1 m S n w w σ = φ σ, (1) a where σ w is the electrical conuctivity of formation water, an a, m, an n are the cementation coefficient, cementation exponent, an saturation exponent, respectively. Density is calculate from the estimate φ an S w accoring to the volume average equation (Smith et al., 2003), ( ) S ( ) ρ = ρ + φ ρ ρ + φ ρ ρ, (2) g HC g w w HC where ρ g an ρ w are ensities of the mineral matrix an formation water, respectively, an ρ HC is either ρ oil or ρ gas. The saturate bulk moulus is calculate from the estimate φ, S w an ry-rock bulk moulus, K, via the Biot-Gassmann flui substitution equation (Mavko et al., 2003), K sat K 1 Ko = K + φ ( 1 φ ) K + K K K 2 fl o o 2, (3) where K o is the bulk moulus of the mineral matrix, an K fl is the bulk moulus of the pore flui compute using Ruess average (Smith et al., 2003), namely, K = S K + ( 1 S fl w w w) K, (4) HC where K w is the bulk moulus of the pore-filling water, an K HC is the bulk moulus of the pore-filling hyrocarbon. Sonic velocities are then calculate from the estimate ensity an ry-rock bulk an shear mouli, K an μ, respectively, using the expressions (Mavko et al., 2003) 3 K + μ sat V = 4, p ρ an (5) V = μ s ρ, (6) where V p is compressional wave velocity, an V s is shear wave velocity. Inverse problem. We use the Levenberg-Marguart metho to minimize the cost function, o 2 C ( m) = m ( ), (7) 2 where o are the measurements, (m) are the synthetic ata, an m are the moel parameters,

3 SPE [ S, S,, S,,log K,log ] T m = w1 w2 w7 φ μ. (8) The ata vector is given by T = α fl log fl, αst log St, α r log r, (9) where fl an Sl are the flexural an Stoneley ispersion ata, respectively, an r are the conuctivity ata. The coefficients α fl, α St, an α r are scaling factors that effectively weight the ata. These factors are chosen to equally scale the Jacobian of each ata type with respect to the water saturation profile. Figure 1 shows Jacobians compute for the gas-bearing sanstone having uniform water saturation of 0.5. The figuresescribe the raial length of sensitivity of each ata type to raial variations of water saturation. As shown in Equations 8 an 9, we use a logarithm transformation to escribe the unknown elastic mouli parameters an the ata. Fig. 1. Sensitvities of (a) flexural ispersion, (b) Stoneley ispersion, an (c) resistivities to perturbations in water saturation at locations extening raially away from the borehole. The backgroun moel is a gas-bearing sanstone with Sw=0.5. Numerical Examples In this stuy, we examine the application of the joint inversion metho to synthetic ata simulate for a clean (clay-free) sanstone, where the sanstone is gas-bearing an invae by WBM filtrate or is water-bearing an invae by OBM filtrate Table 1 lists the assume values of ensity, bulk, an shear mouli of quartz, the mineral matrix of sanstone. Table 2 lists the assume values of ensity an bulk moulus for the water, oil, an gas filling the pore space. Table 1. Assume properties of quartz. Density (g/cm 3 ) 2.65 Bulk moulus, K GPa 37 Shear moulus, μ GPa 44 Table 2. Assume properties of water, oil, an gas. Property Water Oil Gas Density (g/cm 3 ) Bulk moulus (GPa) The formation water conuctivity is assume to be σ w =10 S/m, an Archie s constants are assume to be a=1, an m=n=2. Dry-rock elastic mouli are calculate using the porosity relationships (Mavko et al., 2003), φ K = K 1 o, φc an (10) φ μ = μ 1 o, (11) φc where φ c is the critical porosity, an for sanstones, φ c =40% (Mavko et al., 2003). To moel a slow sanstone, we choose φ c =37.5%. Table 3 lists the resulting ry elastic mouli of the slow sanstone formation consiere in the stuy. Table 3. Assume properties of the slow sanstone formation consiere in this stuy. Porosity Dry bulk moulus, K GPa Dry shear moulus, μ GPa 2.75

4 4 SPE The assume properties of the borehole an borehole flui are: 0.11 m raius; conuctivities of 2.5 S/m for WBM an 10-4 S/m for OBM; ensity of 1 g/cm 3 ; an a compressional velocity of 1500 m/s. Inversion results. Inversion results for the case of the gas-bearing san with presence of WBM invasion are escribe in Table 4 an Figure 2 for noise-free measurements, an Table 5 an Figure 3 for measurements with 0.5% noise ae to the flexural an Stoneley ispersion ata an 1% noise ae to the raw conuctivity ata. Results show that the water saturation profile, as a result of mu-filtrate invasion, is accurately escribe. Moreover, the inversion reners accurate estimates of porosity an ry-rock bulk an shear mouli. Inversion results for the case of the water-bearing san with presence of OBM invasion are escribe in Table 6 an Figure 4 for noise-free measurements, an Table 7 an Figure 5 for measurements with 0.5% noise ae to the flexural an Stoneley ispersion ata an 1% noise ae to the raw conuctivity ata. Results show that the water saturation profile, as a result of mu-filtrate invasion, is accurately inferre. We also obtaine accurate estimates of porosity an ry-rock bulk an shear mouli. To appraise the effectiveness of combining sonic an resistivity measurements in the joint inversion scheme, we perform separate sonic an resistivity inversions for the case of the water-bearing san with presence of OBM invasion, where measurements inclue aitive noise. Table 8 an Figure 6 show results from the inversion performe only with resistivity measurements. The saturation (invasion) profile is poorly escribe, where the uninvae water saturation is greatly uner estimate, an porosity is significantly over estimate. Table 9 an Figure 7 show results from the inversion performe with only sonic measurements. Again, the saturation (invasion) profile is poorly escribe, where the uninvae water saturation is uner estimate. Porosity is uner estimate, ry bulk moulus is over estimate, an the ry shear moulus is etermine with reasonable accuracy. Table 4. Inversion results for the slow, gas-bearing san with presence of WBM invasion. Porosity Dry bulk moulus, Kry GPa Dry shear moulus, μry GPa Table 5. Inversion results for the slow, gas-bearing san with presence of WBM invasion. Resistivity ata have 1% noise, an sonic ata have 0.5% noise. Porosity Dry bulk moulus, Kry GPa Dry shear moulus, μry GPa Table 6. Inversion results for the slow, water-bearing san with presence of OBM invasion. Porosity Dry bulk moulus, Kry GPa Dry shear moulus, μry GPa Table 7. Inversion results for the slow, water-bearing san with presence of OBM invasion. Resistivity ata have 1% noise, an sonic ata have 0.5% noise. Porosity Dry bulk moulus, Kry GPa Dry shear moulus, μry GPa 2.752

5 SPE () Fig. 2. Inversion results for the slow, gas-bearing san with presence of WBM invasion. Sonic an resistivity ata are noise-free. (a) Sonic flexural an Stoneley ispersion ata for the true, initial an inverte moels. (b) Resistivity ata for the true, initial an inverte moels. (c) True, initial, an inverte saturation moel. () Evolution of parameter values versus iteration. (e) Normalize ata misfit versus iteration. (e) () Fig. 3. Inversion results for the slow, gas-bearing san with presence of WBM invasion. Sonic ata inclue 0.5% noise an resistivity ata inclue 1% noise. (a) Sonic flexural an Stoneley ispersion ata for the true, initial an inverte moels. (b) Resistivity ata for the true, initial an inverte moels. (c) True, initial, an inverte saturation moel. () Evolution of parameter values versus iteration. (e) Normalize ata misfit versus iteration. (e)

6 6 SPE () Fig. 4. Inversion results for the slow, water-bearing san with presence of OBM invasion. Sonic an resistivity ata are noise-free. (a) Sonic flexural an Stoneley ispersion ata for the true, initial an inverte moels. (b) Resistivity ata for the true, initial an inverte moels. (c) True, initial, an inverte saturation moel. () Evolution of parameter values versus iteration. (e) Normalize ata misfit versus iteration. (e) () Fig. 5. Inversion results for the slow, water-bearing san with presence of OBM invasion. Sonic ata inclue 0.5% noise an resistivity ata inclue 1% noise. (a) Sonic flexural an Stoneley ispersion ata for the true, initial an inverte moels. (b) Resistivity ata for the true, initial an inverte moels. (c) True, initial, an inverte saturation moel. () Evolution of parameter values versus iteration. (e) Normalize ata misfit versus iteration. (e)

7 SPE Table 8. Inversion results for the slow, water-bearing san with presence of OBM invasion. Separate inversion on resistivity alone. Resistivity ata have 1% noise. Porosity (a) Fig. 6. Inversion results for the slow, water-bearing san in the presence of OBM invasion. Separate inversion on resistivity alone. Resistivity ata inclue 1% noise. (a) Resistivity ata for the true, initial an inverte moels. (b) True, initial, an inverte saturation moel. (b) Table 9. Inversion results for the slow, water-bearing san with presence of OBM invasion. Separate inversion on sonic alone. Sonic ata have 0.5% noise. Porosity Dry bulk moulus, Kry GPa Dry shear moulus, μry GPa (a) Fig. 7. Inversion results for the slow, water-bearing san in the presence of OBM invasion. Separate inversion on sonic ata alone. Sonic ata inclue 0.5% noise. (a) Sonic flexural an Stoneley ispersion ata for the true, initial an inverte moels. (b) True, initial, an inverte saturation moel. (b)

8 8 SPE Conclusions We evelope a new metho for the simultaneous estimation of water saturation, porosity, an ry-rock bulk an shear mouli. These estimates are obtaine through the joint inversion of sonic flexural an Stoneley frequency-ispersion measurements an array-inuction apparent conuctivity measurements. Furthermore, petrophysical an elastic properties are obtaine from measurements acquire in the presence of a raial profile of mu-filtrate invasion. Application of the joint inversion to noisy synthetic ata inicates that the metho provies reliable estimates of the invasion profile (raial profile of S w ), porosity, an ry-rock bulk an shear mouli. Moreover, unesirable parameter estimates obtaine from separate sonic an resistivity inversions confirms the success of the joint inversion approach, an furthermore, inicates that combining measurements that obey ifferent physical principles is avantageous for the petrophysical assessment of porous an permeable formations. Nomenclature a : Archie s tortuosity factor K sat : Saturate bulk moulus [GPa] m : Archie s cementation exponent K : Dry-rock bulk moulus [GPa] n : Archie s saturation exponent K o : Matrix bulk moulus [GPa] σ : Electrical conuctivity [S/m] K fl : Flui bulk moulus [GPa] σ w : Electrical conuctivity of formation water [S/m] K HC : Hyrocarbon bulk moulus [GPa] φ : Porosity [frac.] K w : Formation water bulk moulus [GPa] φ c : Critical porosity [frac.] μ : Dry-rock shear moulus [GPa] S w : Water saturation [frac.] V p : Compressional wave velocity [m/s] ρ : Density, [g/cm 3 ] V s : Shear wave velocity [m/s] ρ w : Formation water ensity, [g/cm 3 ] m : Vector of moel parameters ρ g : Matrix ensity, [g/cm 3 ] C(m) : Cost function ρ HC : Hyrocarbon ensity, [g/cm 3 ] : Vector of ata ρ oil : Oil ensity, [g/cm 3 ] fl : Vector of flexural ispersion ata ρ gas : Gas ensity, [g/cm 3 ] St : Vector of Stoneley ispersion ata r : Vector of conuctivity ata Acronyms AIT : Array-Inuction Tool OBM : Oil-Base Mu WBM : Water-Base Mu 1D : One Dimensional Acknowlegements The work reporte in this paper was fune by the University of Texas at Austin s Research Consortium on Formation Evaluation, jointly sponsore by Anaarko, Aramco, Baker Hughes, BG, BHP Billiton, BP, ConocoPhillips, Chevron, ENI, ExxonMobil, Halliburton, Hess, Marathon, Mexican Institute for Petroleum, Nexen, Petrobras, RWE, Schlumberger, StatoilHyro, TOTAL, an Weatherfor. References Chi, S., Torres-Verín, C., Wu, J, an Alpak, F. O., 2004, Assessment of mu-filtrate invasion effects on borehole acoustic logs an raial profiling of formation elastic properties. In Annual Technical Conference an Exhibition, SPE 90159: Society of Petroleum Engineers. Mavko, G. T., Mukerji, T., an Dvorkin, J., 2003, The rock physics hanbook, tools for seismic analysis in porous meia: Cambrige University Press. Sinha, B. K., Vissapragaa, B., Renlie, L., an Tysse, S., 2006, Raial profiling of the three formation shear mouli an its application to well completions: Geophysics, 71(6), E65-E77. Smith, T., M., Sonergel, C. H., an Rai, C. S., 2003, Gassman flui substitution, a tutorial: Geophysics, 68, Wang, G. L., Torres-Verín, C., Ma, J., an Oumosu, T. B., 2007, Combine inversion of borehole resistivity an sonic measurements to estimate water saturation, porosity, an ry-rock elastic mouli in the presence of invasion. In Proceeings of the SPWLA 48 th Annual Logging Symposium, Austin, Texas, USA.