Assessment of Mud-Filtrate-Invasion Effects on Borehole Acoustic Logs and Radial Profiling of Formation Elastic Properties

Transcription

1 Assessment of Mud-Filtrate-Invasion Effects on Borehole Acoustic Logs and Radial Profiling of Formation Elastic Properties Shihong Chi,* SPE, Carlos Torres-Verdín, SPE, Jianghui Wu,** SPE, and Faruk O. Alpak, SPE, U. of Texas at Austin Summary Despite continued improvements in acoustic-logging technology, sonic logs processed with industry-standard methods often remain affected by formation damage and mud-filtrate invasion. Quantitative understanding of the process of mud-filtrate invasion is necessary to identity and assess biases in the standard estimates of in-situ compressional- and shear-wave (P- and S-wave) velocities. We describe a systematic approach to quantify the effects of mudfiltrate invasion on borehole acoustic logs and introduce a new algorithm to estimate radial distributions of elastic properties away from the borehole wall. Radial saturation distributions of mud filtrate and connate formation fluids are obtained by simulating the process of mud-filtrate invasion. Subsequently, we calculate radial distributions of the elastic properties using the Biot-Gassmann fluid-substitution model. The calculated radial distributions of formation elastic properties are used to simulate array sonic waveforms. Finally, estimated P- and S-wave velocities for homogeneous, stepwise, and multilayered formation models are compared to quantify mud-filtrate-invasion effects on sonic measurements. We use a nonlinear Gauss-Newton inversion algorithm to estimate radial distributions of formation elastic parameters in the presence of invaded zones using normalized spectral ratios of array waveform data. Inversion examples using synthetic and field data indicate that physically consistent distributions of formation elastic properties can be reconstructed from array waveform data. In turn, radial distributions of formation elastic properties can be used to construct more-realistic near-wellbore petrophysical models for applications in reservoir simulation and production. Introduction During and after drilling, the near-wellbore formation is often altered by stress buildup and release, mud-filtrate invasion, chemical reactions, and many other factors. These alterations cause the physical properties in the near-wellbore region to be different from those of the virgin rock formation. Stress concentration around a wellbore may cause near-wellbore damage and induce formation anisotropy on P- and S-wave velocities. The stress-induced anisotropy can be identified by dispersion analysis (Plona et al. 2002). Positive radial velocity gradients focus the elastic waves propagating away from the wellbore back toward the borehole wall. Such a phenomenon can be identified easily from high-amplitude acoustic arrivals. In this paper, we focus our attention on mudfiltrate-invasion effects only. It is well known that formation properties inferred from wireline logging measurements may not reflect true properties of virgin formations. A realistic description of the invaded zone is important * Currently with ConocoPhillips. ** Currently with Baker Atlas. Currently with Shell Intl. E&P. Copyright 2006 Society of Petroleum Engineers This paper (SPE 90159) was first presented at the 2004 SPE Annual Technical Conference and Exhibition, Houston, September, and revised for publication. Original manuscript received for review 4 June Revised manuscript received 29 May Paper peer approved 25 July for the processing and interpretation of sonic logs. A common model used in the open literature assumes that a sharp interface exists between the altered zone and the undisturbed formation (Baker 1984). The term stepwise is used to describe this type of mud-filtrate-invasion model. Linear-gradient models have been described for syntheses of acoustic waveforms as well (Stephen et al. 1985). Actual radial distributions of elastic wave properties resulting from invasion can be complex and are dependent on the specific petrophysical properties of the rock as well as on the static and dynamic properties of the fluids involved. We divide the invaded zone into a set of concentric radial layers to represent an actual invasion profile and call it a multilayered model. Subsequently, we describe a procedure for calculating the radial distributions of formation elastic properties with the Biot-Gassmann fluid-substitution model starting from numerically simulated radial distributions of flow saturation. Theoretical and experimental studies (Geertsma and Smit 1961; Biot 1956a, 1956b; Toksöz et al. 1976; Domenico 1976; Dutta and Ode 1979; Murphy 1982) have shown that rocks saturated with hydrocarbons and water can be differentiated with acoustic velocity measurements. Using time-lapse acoustic logging, bypassed zones can be identified, and fluid movement in rock formations can be monitored in open and cased holes. If the invasion depth is beyond the radial length of investigation of a logging tool, the measured velocities will reflect those of the invaded zone, and log corrections will be necessary to estimate virgin-formation velocities. The current industry practice of acoustic-data processing makes use of the slowness-time coherence (STC) method to estimate in-situ velocities of rock formations. This type of processing yields values of P- and S-wave velocities, both of which are intricate functions of formation and borehole properties, frequency, and number of time samples used in the time window of the STC algorithm. When invasion into a formation is significant, the STC method still gives one P-wave velocity and/or S-wave velocity at each depth of measurement, which may reflect only properties of the invaded zone. However, waveforms recorded by modern acoustic tools may contain valuable information on the radial distribution of near-wellbore formation properties. In an effort to estimate radial distributions of near-wellbore formation properties, Hornby (1993) developed a tomographic reconstruction technique that yielded formation P-wave velocities from sonic travel times. The estimation was performed with a series of linear inversions followed by ray tracing. However, travel-time tomography using ray tracing requires accurate picking of the refracted arrivals of each wave train and the use of an accurate high-frequency ray-trace approximation. Few studies of seismic full-waveform inversion (Sen and Stoffa 1991; Zhou et al. 1997; Pratt 1999a, 1999b) show that the use of amplitude data can improve the resolution of 1D vertical distributions of velocity and density. The amplitude and phase behavior of full-waveform data are sensitive to the petrophysical properties of formations supporting the propagation of elastic waves. Therefore, full-waveform analysis has significant potential in acoustic logging to estimate petrophysical properties of invaded zones and virgin formations. The reconstruction of formation slowness away October 2006 SPE Reservoir Evaluation & Engineering 553

2 from the borehole wall can provide valuable calibration factors for measurements acquired with shallow-reading devices. Corrections provided by full-waveform inversion may be of primary importance for the robust and accurate computation of synthetic seismograms. There is, however, one major difficulty to overcome in full-waveform inversion. In all field applications, the effective source wavelet; the coupling among source, borehole, and formation; and the coupling between receivers and the formation depend on the in-situ borehole conditions. The practical difficulty in estimating the sonic-source signature is probably the reason why very few attempts have been made to use the information borne by the full waveform of acoustic-logging data. Cheng (1989) described an indirect method for determining S-wave velocities from full-waveform acoustic logs by means of inversion. This method makes use of the spectral ratio of the P-wave trains at two source/receiver separations and simultaneously inverts for both S-wave velocity and P-wave attenuation. In that paper, Cheng s approach based on spectral-ratio data was extended to full-waveform arrays to overcome the practical difficulty of the unknown source-output spectrum. Lee and Kim (2003) estimated P-wave velocity distributions from synthetic seismic data by use of a spectral-ratio method. Frazer et al. (1997) and Frazer and Sun (1998) described an inversion method for processing array full-waveform data. Their approach does not require an exact source function, although it is a necessary part of the inversion. Because of this, the chosen source function can bias the inversion results. Our inversion approach first transforms array waveforms into the frequency domain to construct a set of normalized array sonic data. The normalized wavefield is independent of the spectrum of the source; hence, the proposed inversion method allows one to make use of the full-waveform content of sonic data without requiring knowledge of the source signature. Previous studies of full-waveform borehole acoustic measurements assumed a homogeneous and isotropic formation and a model for borehole wave propagation without surface irregularities. In this paper, we make the assumptions of a radially multilayered formation model and a cylindrical borehole. Methodology Numerical Simulation of Mud-Filtrate Invasion. Mud-filtrate invasion is regarded as a water- or oil-injection process into a gas or oil reservoir wherein two-phase immiscible fluid flow is assumed in the simulations. To study the effects of different types of mud filtration on gas or oil reservoirs, the following cases are selected: (a) water-based mud invades oil and gas reservoirs penetrated by open holes and (b) oil-based mud invades gas reservoirs penetrated by open holes. In each of these cases, it is assumed that a vertical well penetrates horizontally layered rock formations and that the rock formation considered does not communicate hydraulically with its upper and lower shoulder beds. To study the sensitivity of mud-filtrate invasion to formation petrophysical properties, we consider a low-permeability (30 md) and low-porosity (15 p.u.) reservoir and a high-permeability (300 md) and highporosity (30 p.u.) reservoir for Cases (a) and (b). Details of the numerical simulation of the process of mudfiltrate invasion in open holes can be found in Wu et al. (2004, 2005) and George et al. (2004). In the simulations, both the dynamic growth of the mudcake and the dynamic decrease of the mudcake permeability are coupled to formation properties. This results in a dynamic monotonic decrease of flow rate into the formation. After a short initial spurt of mud-filtrate invasion, the invasion process reaches a steady state. Cycles of mudcake ruboff and buildup can also affect the process of mud-filtrate invasion. Fig. 1 is a schematic of the process of mud filtrate invading a permeable rock formation. The invasion front moves deeper into the formation with time. Fig. 2 shows the time evolution of waterbased mud-filtrate saturation in the radial direction of a lowporosity, low-permeability, and oil-bearing sandstone formation. The water-based mud filtrate reaches a radial distance of approximately 0.8 m after 4 days of invasion. Mud-filtrate invasion is shallower for the corresponding high-porosity case. Fig. 3 shows Fig. 1 Schematic of the process of mud-filtrate invasion in rock formations penetrated by a mud-filled and overbalanced borehole. that mud filtrate penetrates to a radial distance of 0.5 m after 4 days of invasion. Because of relative permeability, in the example case considered here, oil-based mud invades the gas-bearing sandstone formation at a much slower rate than for the case of water-based mud-filtrate invasion. Fig. 4 shows the time evolution of oil-based mud-filtrate saturation in the radial direction of a low-porosity (15 p.u.) and low-permeability (30 md) formation. Oil-based mud filtrate reaches a radial distance of approximately 0.2 m after 4 days of invasion. Fig. 5 shows that mud filtrate primarily concentrates within a distance of 0.1 m away from the borehole wall. Radial Distributions of Density and P- and S-Wave Velocities in an Invaded Zone. Once radial profiles of invasion are obtained from numerical simulations, radial distributions of the elastic properties of a formation can be calculated using the Biot-Gassmann fluid-substitution equations. Following the tutorial presented by Smith et al. (2003), a calculation is first performed for the basic formation and fluid properties. Subsequently, for each point in the saturation profile we calculate (a) fluid properties, (b) saturated bulk modulus of the rock, (c) bulk density of the rock, and (d) P- and S-wave velocities. When computing fluid and formation properties after invasion, a careful selection of homogeneous or patchy saturation models is necessary. Over geologic time, fluids in the pores of the rocks become homogeneously distributed. This is one of the important Fig. 2 Time evolution of mud-filtrate invasion for the case of water-based mud invading a 15%-porosity, oil-bearing rock formation. From left to right, the first curve is the water-saturation profile 1 day after the onset of mud-filtrate invasion. The time increment is 1 day, and the rightmost curve describes water saturation at the 16th day. 554 October 2006 SPE Reservoir Evaluation & Engineering

3 Fig. 3 Time evolution of mud-filtrate invasion for the case of water-based mud invading a 30%-porosity, oil-bearing reservoir. From left to right, the first curve is the water-saturation profile 1 day after the onset of mud-filtrate invasion. The time increment is 1 day, and the rightmost curve describes water saturation at the 16th day. assumptions of the Biot-Gassmann fluid-substitution model. However, mud-filtrate invasion in the near-wellbore region changes the connate-fluid distribution. From the numerical-simulation results described in this paper, it is clear that the equilibrium of fluid phases in the near-wellbore region is not established within a few days after the onset of the mud-filtrate-invasion process. Therefore, it is more appropriate to use a patchy saturation model to compute the corresponding bulk modulus (Hill 1963). Numerical Simulation of Time-Lapse Borehole Acoustic Measurements. On the basis of saturation-profile and fluidsubstitution calculations, we obtain a discretized and concentric multilayered rock model. The radial discretization is sufficient to ensure accurate representation of the continuous radial distributions in the calculation of elastic properties. A corresponding stepwise-invaded-zone model also is used for the numerical simulation. The waveforms simulated for the multilayered and stepwise models are compared, with the objective of determining when the stepwise models are adequate to represent the invaded zone. Fig. 4 Time evolution of mud-filtrate invasion for the case of oil-based mud invading a 15%-porosity, gas-bearing reservoir. From left to right, the first curve is the water saturation profile 1 day after the onset of mud-filtrate invasion. The time increment is 1 day, and the rightmost curve describes oil saturation at the 16th day. Fig. 5 Time evolution of mud-filtrate invasion for the case of oil-based mud invading a 30%-porosity, gas-bearing reservoir. From left to right, the first curve is the water saturation profile 1 day after the onset of mud-filtrate invasion. The time increment is 1 day, and the rightmost curve describes oil saturation at the 16th day. In the synthesis of array waveforms, we assume a representative source/receiver configuration. For monopole logging, the central frequency of the source is 10 khz. The minimum source/ receiver distance is 9 ft, with the receiver array consisting of eight receivers spaced at 6-in. intervals. Dipole logging makes use of a 1.5-kHz source and a minimum source/receiver separation of 11.5 ft. In the lower dipole mode, the eight source/receiver offsets vary from 3.51 m (11.5 ft) to 4.57 m (15 ft) with a receiver spacing of 0.15 m (6 in.). We simulate sonic array waveforms for a homogeneous virgin formation as well as for a saturated formation after 4 days of mud-filtrate invasion. Processing and Interpretation of Monopole and Dipole Logging Data. The sensitivity of acoustic measurements to different invasion models and radial saturation profiles can be assessed by comparing the synthetic waveforms for homogeneous, stepwise, and multilayered radial formation models. Arrival times and amplitudes of P- and S- wave modes are the main waveform features used to assess the effect of mud-filtrate invasion on borehole acoustic logging measurements. We use an industry standard STC method (Kimball and Marzetta 1984) to determine the P- and S-wave velocities of array sonic waveforms. Radial depths of investigation for sonic tools depend on formation elastic properties, transmitter-to-receiver spacing, wavelength and wave mode considered, and frequency, among other factors. Because we chose a typical sonic-tool configuration and source frequencies used by the logging industry, we can focus our study of invasion effects on velocity measurements. After P- and S-wave velocities are determined, the depth of investigation can be ascertained from the inspection of radial velocity profiles. When correlating seismic data with acoustic logs, it is often found that synthetic seismograms do not match the measured seismograms. As a result, corrections to the acoustic logs by means of Biot-Gassmann fluid substitution are common in the seismic industry to eliminate mud-filtrate-invasion effects from sonic logs. In this correction procedure, it is assumed that the measured velocities are those of the invaded zone saturated with mud filtrate. For the case of oil-based mud, filtrate mixing with gas may have to be considered before fluid substitution. By displacing the saturation fluid in the invaded zone with the connate formation fluid and by applying the Biot-Gassmann fluid-substitution equation, new velocities are obtained and taken as virgin-formation velocities. We investigate the validity of this practice through several case studies. October 2006 SPE Reservoir Evaluation & Engineering 555

4 Full-Waveform Inversion. In our inversion of radial elastic properties, we assume a cylindrical borehole surrounded by concentric multilayered formations. We first transform array sonic waveforms into the frequency domain to construct a set of normalized array sonic spectra. The normalized wavefield is independent of the spectrum of the source; hence, the proposed inversion method allows one to make use of the full-waveform content of sonic data without requiring knowledge of the source signature. The normalized wavefield (or spectrum) of full waveforms constitutes the measurement data input to the full-waveform inversion algorithm. For estimation of the radial distributions of elastic properties, density and P- and S-wave velocities are assigned to each concentric layer in the near-wellbore region. Subsequently, we simulate the measured wavefields with the assumed properties. A cost function that enforces the quadratic difference between the simulated and measured normalized wavefields is used for inversion. We make use of a Gauss-Newton fixed-point iteration search to determine a stationary point at which the cost function attains a minimum. The stationary point yields the radial distribution of elastic properties. Details of the full-waveform inversion are given in the Appendix. Case Studies Sandstone Oil Reservoir Invaded With a Water-Based Mud. In onshore drilling activities, water-based mud is still widely used because of its efficiency to balance formation pressure without substantially increasing costs. Elastic properties of oil in rock formations are much closer to those of mud filtrate derived from a water-based mud. Compared to the case of mud filtrate invading a gas reservoir, it may be more difficult to distinguish mud-filtrateinvasion effects on the measured velocities. Moreover, fluid-flow properties of oil and gas are very different from each other. To study the sensitivity of the measured velocities to porosity and permeability, this section first considers a low-porosity (15 p.u.) and low-permeability (30 md) reservoir. Table 1 summarizes the petrophysical and fluid properties used in the simulation of waterbased mud-filtrate invasion. Subsequently, we focus our attention to the case of a high-porosity (30 p.u.) and high-permeability (300 md) reservoir. Saturation profiles of mud filtrate and oil in formations are simulated for 4 days of invasion. The latter profiles are used to calculate radial distributions of elastic properties using the Biot-Gassmann fluid-substitution model. Fig. 2 shows the time evolution of mud-filtrate invasion for the case of water-based mud invading the oil-bearing sandstone reservoir of 15% porosity. Fig. 6 shows the corresponding radial distributions of density and P- and S-wave velocities calculated with the fluid-substitution parameters described in Table 2. As shown in Fig. 6, in the invaded zones the P-wave velocity is higher than that of the virgin formation, while the S-wave velocity becomes lower than the original velocity. P-wave velocities estimated from the array waveforms (Fig. 7a) simulated for stepwise and multilayered models are approximately 1% lower than those of the virgin formation. STC processing can introduce 1% uncertainty in the estimation of P-wave velocity. Therefore, the 1% lower P-wave velocity may result from STC processing. This implies that the P-wave is not sensitive to the invaded zone, which exhibits a higher P-wave velocity. On the other hand, S-wave velocities obtained from STC processing of monopole and dipole waveforms are 2% lower than those of the virgin formation. This behavior agrees with the velocity decrease described in Fig. 6 and indicates that S-wave propagation is sensitive to the invaded zone. Fig. 8 shows the radial distributions of density and P- and S-wave velocities for the high-porosity and high-permeability case after 4 days of invasion. The corresponding STC results show that the P-wave velocity is the same as that of the virgin formation within the processing error bound of 1%. This behavior indicates that P-wave propagation is sensitive to the radial region beyond the invaded zone. The S-wave velocity obtained for this invaded model is approximately 2.6% lower than that of the virgin zone. This relatively larger reduction in S-wave velocity compared to the low-porosity case is expected because more oil is replaced by water-based-mud filtrate. In summary, P-wave propagation is sensitive to the virgin zone, whereas S-wave propagation is sensitive to the invaded zone when the radial length of mud-filtrate invasion is approximately 0.5 to 0.8 m for the two sandstone reservoirs considered in this section. Sandstone Gas Reservoir Invaded With an Oil-Based Mud. Our study indicates that, in general, oil-based mud causes shallower invasion than water-based mud and, hence, less formation damage. Because oil-based mud is also chemically less active compared to water-based mud, it is more effective in inhibiting shale swelling. This is one of the reasons why oil-based mud is widely used to drill expensive offshore wells. Fig. 6 Radial distributions of density and P- and S-wave velocities for the case of water-based mud invading a 15%-porosity, oil-bearing reservoir after 4 days of invasion. 556 October 2006 SPE Reservoir Evaluation & Engineering

5 We selected two synthetic cases to study mud-filtrate effects on the measured velocities of sandstone formations exhibiting low and high porosities and permeabilities. Fig. 9 shows the radial distributions of density and P- and S-wave velocities for the case of a gas-bearing sandstone reservoir of 15% porosity invaded with an oil-based mud. P-wave velocities estimated from array waveforms simulated for a stepwise and a multilayered model are approximately 1.0% higher than that of the true formation P-wave velocity, but approximately 3.0% lower than that of an invaded zone saturated with 90% mud filtrate. Estimated S-wave velocities are approximately 0.2% lower than that of the true formation S-wave velocity, while the S-wave velocity for the invaded zone is 1.2% lower than that of the true formation. Therefore, P- and S-wave velocities are slightly influenced by mud-filtrate invasion and remain primarily sensitive to the radial region beyond the invaded zone. For the 30%-porosity case, Fig. 10 shows the radial distributions of density and P- and S-wave velocities. It is observed from the monopole waveforms (Fig. 11) that the presence of invaded zones does not affect the P-wave arrival time, but it does delay the S-wave arrival time. On the other hand, P-wave amplitudes do not change appreciably, whereas S-wave amplitudes increase. In the dipole waveforms, the change in S-wave arrival is negligible. Dipole waveforms for the stepwise model are significantly different from those simulated for the multilayered formation model. P-wave velocities estimated from data for homogeneous, multilayered, and stepwise models are the same. This behavior indi- Fig. 7 Simulated (a) monopole and (b) dipole waveforms for the homogeneous, stepwise, and multilayered formation models shown in Fig. 6. Fig. 8 Radial distributions of density and P- and S-wave velocities for the case of water-based mud invading a 30%-porosity, oil-bearing rock formation after 4 days of invasion. Fig. 9 Radial distributions of density and P- and S-wave velocities for the case of oil-based mud invading a 15%-porosity, gas-bearing rock formation after 4 days of invasion. October 2006 SPE Reservoir Evaluation & Engineering 557

6 Fig. 10 Radial distributions of density and P- and S-wave velocities for the case of oil-based mud invading a 30%-porosity, gas-bearing rock formation after 4 days of invasion. cates that P-wave propagation is sensitive to the radial region past the invaded zone. S-wave velocities estimated for the stepwise and the multilayered models are approximately 1% lower than that of the virgin formation. According to the Biot-Gassmann fluidsubstitution model, a rock formation saturated with 90% mud filtrate should exhibit a 4.3% S-wave velocity decrease compared to that of the virgin formation. This indicates that the measured S-wave velocities are primarily sensitive to the virgin formation but remain influenced by mud filtrate in the near-wellbore region. In all the cases studied in this section, both P- and S-waves are not sensitive to the invaded zone when the oil-based mudfiltrate invasion reaches a radial length of approximately 0.3 m in sandstone reservoirs. Elastic Radial Profiles of a Fast Rock Formation. This inversion case makes use of realistic elastic properties for a fast-formation model (Schmitt 1988), which consists of six radial layers, including a fluid layer in the borehole. Table 3 describes the actual fluid and formation properties and the radial discretization grid used for the inversion. Amplitude spectra of the simulated waveforms are used in the estimation to accelerate the convergence of the inversion algorithm. Fig. 12 shows radial distributions and crossplots of actual and inverted elastic properties after 26 iterations of the inversion algorithm. The inverted radial distributions of elastic properties show a good agreement with the actual properties. Global correlation coefficients calculated between the inverted and the actual values of elastic properties indicate that all the inverted elastic properties exhibit high global similarity with the original model properties (0.932, 1, and 1 for P- and S-wave velocities and density, respectively). Fig. 11 Simulated (a) monopole and (b) dipole waveforms for the case of homogeneous, stepwise, and multilayered formation models invaded with oil-based mud after 4 days of invasion in a 30%-porosity gas reservoir. Inversions performed with no assumption of an increasing value of elastic properties away from the borehole wall converge to local minima without exception. Fig. 13 shows the radial distributions and crossplots of actual and inverted elastic properties after 20 iterations for a slow formation under the assumption of monotonically increasing values of elastic properties. The inversion stops at a local minimum, as can be observed from the evolution of the data misfit and cost functions shown in Figs. 13a and 13b, respectively. Global correlation coefficients for the inverted and actual velocities are relatively high (0.83 and 0.83 for P- and S-wave velocities, respectively). However, for bulk density, the correlation coefficient is This indicates the relatively low sensitivity of the measurements to radial variations of density in Elastic Radial Profiles of a Slow Rock Formation. This example is an extension of the slow formation model in the simple borehole cases described by Cheng (1989) within the context of marine sediments. Table 4 describes the model, which consists of six radial layers, including one fluid layer in the borehole. The tool and source configurations are the same as those discussed previously for simple borehole cases. Given that the low-frequency content of the full waveforms is more important in soft formations than in fast formations, data from the frequency band 2 5 khz are used for inversion. Inversion results from this example not only estimate the virgin-formation elastic properties but also give a description of near-wellbore damage. 558 October 2006 SPE Reservoir Evaluation & Engineering

7 Fig. 12 Simultaneous inversion of radial distributions of elastic properties for a six-layer fast formation using noise-free normalized spectra of array waveform data. Panel (a) shows the array waveform in the time domain, and Panel (b) shows the data residuals yielded by the inversion. In Panel (c), the inverted radial distributions of elastic properties are identified with solid lines and open circles; additional lines identify original model properties. Panel (d) shows that the correlation coefficient, r 2, is the correlation coefficient between the inverted and actual elastic properties. the 2- to 5-kHz frequency band. Overall, the inversion algorithm provides a reliable way to estimate radial distributions of elastic properties in soft rock formations. This is a valuable tool for detailed analysis of acoustic-logging data in offshore wells. Elastic Radial Profiles From Field Data. This example assesses the feasibility and performance of the inversion algorithm when applied to field data. Full-waveform acoustic data were acquired with the Dipole Sonic Imager (DSI)* tool in the depth interval from 13,000 to 13,050 ft within a well penetrating a tightsandstone gas reservoir (Anderson Well No. 2). Core data and log measurements indicate that the porosity of the fine-grained sandstone formation is below 9%. Gas saturation ranges from 80% to 95%. Fig. 14a displays the array waveform data acquired at the depth of 13,030 ft. The bottom four traces show very similar characteristics in the number of wave modes and in the amplitudes of each wave mode. The top four traces, however, exhibit a completely different character. Fig. 14b displays the amplitude spectra of the array waveform data. The main energy of the waveforms is contained in the frequency band from 9 to 14 khz. Input data to the inversion algorithm are chosen to be the normalized frequency data in the band of khz, given that previous examples show that data from a narrower frequency-data band improve the convergence of the algorithm. The STC processing method yields formation P- and S-wave slowness values of and s/ft, respectively, which are used as the initial properties for the inversion. Likewise, the density log reads a value of g/cm 3 for bulk formation density. The mud density at this depth in the * Mark of Schlumberger. October 2006 SPE Reservoir Evaluation & Engineering 559

8 Fig. 13 Radial distributions and crossplots of the actual and inverted elastic properties for a six-layer slow formation. The evolution of the data misfit and cost function with iteration number is shown in Panels (a) and (b), respectively. In Plot (c), the inverted radial distributions of elastic properties are identified with open circles. Additional lines identify original model properties. Panel (d) shows the correlation coefficient, r 2, between the inverted and actual elastic properties. borehole is 14 lbm/gal (1678 kg/m 3 ), whereas the acoustic velocity of the mud is approximately 1186 m/s. Mud-filtrate-invasion studies (Salazar et al. 2005) indicate that mud filtrate reaches a radial distance of approximately 1.5 ft in the flow unit of interest after 24 hours of drilling. Thus, we use a concentric five-radial-layer model to describe the near-wellbore invasion zone. The inner radii of the radial formation layers are 0.07, 0.15, 0.30, 0.40, and 0.45 m, respectively, and the outmost layer is assumed to be unbounded in the radial direction. Within each radial zone, formation elastic properties are assumed constant. First, normalized frequency data from Traces 1 through 4 in the band from 11 to 14 khz are used as input data for the inversion. The data misfit decreases to 4% after 20 iterations. Fig. 14c indicates that the formation P- and S-wave velocities increase and bulk density decreases in the radial direction. It is known that formation damage caused by drilling decreases the formation velocities and that mud-filtrate invasion in the near-wellbore region increases the bulk density. The inverted radial distributions indicate that a damaged zone exists in the near-wellbore region even though the borehole is in excellent condition. Such a result also agrees with the high-amplitude P-wave components that are observed from Traces 1 through 4 of the array sonic waveforms. A radial profile of monotonically increasing P-wave velocity focuses the elastic waves propagating away from the wellbore back toward the borehole wall and increases the P-wave amplitude (Winkler 1997; Chen et al. 1996). The radial distribution of density shows a decreasing trend from the borehole wall into the formation. Such a behavior can be interpreted as due to mud-filtrate displacing gas in the near-wellbore region. This exercise confirms that the inversion algorithm is reliable to estimate radial distributions of formation elastic properties in the near-wellbore region. Conclusions In the cases of water-based mud invading oil-bearing sandstone reservoirs, P-wave propagation remains sensitive to virgin zones, whereas S-wave propagation is affected by invaded zones when the radial length of invasion of mud filtrate is approximately 0.5 to 0.8 m. In this latter situation, log corrections are not necessary for P-wave velocities, whereas detailed saturation-profile calculations are needed to correct the measured S-wave velocities for mudfiltrate-invasion effects. For the cases of oil-based mud invading sandstone reservoirs, both P- and S-wave propagations are insensitive to the presence of mud filtrate in the invaded zones. We developed a new full-waveform inversion algorithm that makes use of the normalized frequency spectra of sonic waveforms. The reliability of the inversion algorithm was tested successfully with radially multilayered 1D synthetic models. In addition, we obtained petrophysically consistent results when the in- 560 October 2006 SPE Reservoir Evaluation & Engineering

9 Fig. 14 Simultaneous inversion of radial distributions of elastic properties using array waveform data acquired in the Anderson Well No. 2 and penetrating a tight gas reservoir. Panel (a) shows the array waveforms in the time domain, and Panel (b) shows the corresponding amplitude spectra. Panel (c) shows the homogeneous-formation model used to initialize the inversion and the inverted radial distributions of formation elastic properties. version algorithm was applied to field data acquired in a tight gas reservoir. Inversion exercises performed on synthetic data indicate that the low-frequency content of full-waveform data is sensitive to formation elastic properties radially away from the borehole wall. Thus, high-frequency components of array sonic data may be preferable for estimating the radial distribution of elastic properties in the near-wellbore region. Nomenclature C(m) cost function d(m) measurement vector numerically simulated for specific values of m d obs measurement vector i index or imaginary unit Im imaginary j, k indices J(m) Jacobian matrix l i lower bound k f fluid wave number k pf radial wave number of fluid m size-n vector of unknown parameters m R size-n reference vector M number of frequency-domain measurements N number of radial layers N FREQ number of frequencies used for each trace N REC number of receivers r 1 radius of the first radial formation layer R borehole radius Rˆ (1) + reflectivity between the fluid and the first radial layer of the formation Re real S j spectra of normalized waveforms S( ) effective-source-output spectrum T transpose T 12 transfer function between the two receivers u i upper bound V f fluid acoustic velocity V p P-wave velocity V s S-wave velocity W d W T d inverse of the measurement covariance matrix W m W T m inverse of the model covariance matrix z 1, z 2 receiver locations prescribed value of enforced data misfit Lagrange multiplier or regularization parameter density October 2006 SPE Reservoir Evaluation & Engineering 561

10 Acknowledgments The work reported in this paper was funded by the U. of Texas at Austin s Research Consortium on Formation Evaluation, jointly sponsored by Aramco, Baker Atlas, BP, British Gas, ConocoPhillips, Chevron, Eni E&P, ExxonMobil, Halliburton Energy Services, Marathon, the Mexican Inst. for Petroleum, Norsk-Hydro, Occidental Petroleum Corp., Petrobras, Schlumberger, Shell Intl. E&P, Statoil, Total, and Weatherford. References Baker, L.J The effect of the invaded zone on full wavetrain acoustic logging. Geophysics 49: DOI: Biot, M.A. 1956a. Theory of propagation of elastic waves in a fluid saturated porous solid I. Low-frequency range. J. Acoust. Soc. Am. 28: DOI: Biot, M.A. 1956b. Theory of propagation of elastic waves in a fluid saturated porous solid II. Higher frequency range: J. Acoust. Soc. Am. 28: DOI: Chen, X., Quan, Y., and Harris, J.M Seismogram synthesis for radially layered media using the generalized reflection/transmission coefficients method: Theory and applications to acoustic logging. Geophysics 61: DOI: Cheng, C.H Full waveform inversion of P waves for V s and Q p. J. of Geophys. Res. 94: Cheng, C.H., Wilkens, R.H., and Meredith, J.A Modeling of full waveform acoustic logs in soft marine sediments. Paper LL presented at the 27th SPWLA Annual Logging Symposium, Houston, 9 13 June. Domenico, S.N Effect of brine-gas mixture on velocity in an unconsolidated sand reservoir. Geophysics 41: DOI: dx.doi.org/ / Dutta, N.C. and Ode, H Attenuation and dispersion of compressional waves in fluid-filled rocks with partial gas saturation, White model: Part I Biot theory. Geophysics 44: DOI: dx.doi.org/ / Frazer, L.N. and Sun, X New objective functions for waveform inversion: Geophysics 63: DOI: Frazer, L.N., Sun, X., and Wilkens, R.H Inversion of sonic waveforms with unknown source and receiver functions. Geophys. J. Int. 129: Geertsma, J. and Smit, D.C Some aspects of elastic wave propagation in fluid-saturated porous solids. Geophysics 26: DOI: George, B.K., Torres-Verdín, C., Delshad, M., Sigal, R., Zouioueche, F., and Anderson, B Assessment of in-situ hydrocarbon saturation in the presence of deep invasion and highly saline connate water. Petrophysics 45 (2): Gill, P.E., Murray, W., and Wright, M.H Practical Optimization. London: Academic Press. Hill, R Elastic properties of reinforced solids: Some theoretical principles. J. Mech. Phys. Solids 11: DOI: / (63)90036-X. Hornby, B.E Tomographic reconstruction of near borehole slowness using refracted borehole sonic arrivals. Geophysics 58: DOI: Kimball, C.V. and Marzetta, T.L Semblance processing of borehole acoustic array data. Geophysics 49: DOI: / Lee, K.H. and Kim, H.J Source-independent full waveform inversion of seismic data. Geophysics online (published electronically on 20 May 2003). DOI: Murphy, W.F. III Effects of microstructure and pore fluids on the acoustic properties of granular sedimentary materials. Ph.D. thesis, Stanford U., Stanford, California. Plona, T., Sinha, B., Kane, M. et al Mechanical Damage Detection and Anisotropy Evaluation Using Dipole Sonic Dispersion Analysis. Paper F presented at the 43rd Annual SPWLA Conference, Osio, Japan, 2 5 June. Pratt, R.G. 1999a. Seismic waveform inversion in frequency domain, Part 1: Theory and verification in physical scale model. Geophysics 64: DOI: Pratt, R.G. 1999b. Seismic waveform inversion in frequency domain, Part 2: Fault delineation in sediments using crosshole data. Geophysics 64: DOI: Salazar, J.M., Torres-Verdín, C., and Sigal, R Assessment of permeability from well logs based on core calibration and simulation of mud-filtrate invasion. Petrophysics 46 (6): Schmitt, D.P Shear wave logging in elastic formations. J. Acoust. Soc. of Am. 84: DOI: Sen, M.K. and Stoffa, P.L Nonlinear one-dimensional seismic waveform inversion using simulated annealing. Geophysics 56: DOI: Smith, T.M., Sondergeld, C.H., and Rai, C.S Gassmann fluid substitutions: A tutorial. Geophysics 68: DOI: / Stephen, R.A., Cardo-Casas, F., and Cheng, C.H Finite difference synthetic acoustic logs. Geophysics 50: DOI: dx.doi.org/ / Toksöz. M.N., Cheng, C.H., and Timur, A Velocities of seismic waves in porous rocks: Geophysics 41: DOI: dx.doi.org/ / Toksöz, M.N., Wilkens, R.H., and Cheng, C.H Shear wave velocity and attenuation in ocean bottom sediments from acoustic log waveforms. Geophys. Res. Lett. 12: Torres-Verdín, C. and Habashy, T.M Rapid 2.5-dimensional forward modeling and inversion via a new nonlinear scattering approximation. Radio Science 29 (4): DOI: /94RS Winkler, K.W Acoustic evidence of mechanical damage surrounding stressed boreholes. Geophysics 62: DOI: / Wu, J., Torres-Verdín, C., Sepehrnoori, K., and Proett, M.A The influence of water-base mud properties and petrophysical parameters on mudcake growth, filtrate invasion, and formation pressure. Petrophysics 46 (1): Wu, J., Torres-Verdín, C., Sepehrnoori, K., and Delshad, M Numerical Simulation of Mud-Filtrate Invasion in Deviated Wells. SPEREE 7 (2): SPE PA. DOI: /87919-PA. Zhou, C., Schuster, G.T., Hassanzadeh, S., and Harris, J.M Elastic wave equation travel time and wavefield inversion of crosswell data. Geophysics 62: DOI: Appendix Methodology for Full-Waveform Inversion The model of wave propagation in a borehole considered in this paper assumes an isotropic, radially multilayered formation with no geometrical irregularities along the borehole wall. As with any inversion procedure, the accuracy and reliability of the results will depend on how accurately the forward model describes actual in-situ conditions. In practical field studies, irregularities on the borehole wall can be quantified with caliper measurements. Consequently, vertical formation intervals can be identified that are consistent with the assumption of a smooth borehole wall. Along the length of the receiver spacing (6 in.), the formation can be regarded as homogeneous in the axial direction. In practice, the effect of borehole/receiver coupling can be assumed negligibly small in comparison to source coupling. The spectral ratio of the pressure responses at two receiver locations, z 1 and z 2, from a common source can be written as T 12 = S i S i Rˆ 1 + e ik p r f 1 e ik e ikz f z 2 1 Rˆ 1 + e ik 2 p r dk + f 1 z 2 Rˆ 1 + e ik p r f 1 e ik e ikz f z 1 1 Rˆ 1 + e ik 1 p r dk + f 1 z 1,...(A-1) where S( ) is an effective-source-output spectrum, k f is the (1) fluid wave number, k pf is the radial wave number of fluid, Rˆ + is related to the reflectivity between the fluid and the first ra- 562 October 2006 SPE Reservoir Evaluation & Engineering

11 dial layer of formation, and r 1 is the radius of the first layer. The ratio of the two source terms in Eq. A-1 is equal to unity, resulting in a final frequency representation independent of the source spectrum. This procedure indicates that, in principle, knowledge of the source spectrum is not necessary for full-waveform inversion when spectral ratios are used as input data. We note that the term T 12 in Eq. A-1 is the transfer function between the two receivers. For practical applications, the source functions are never fully known because of the variability of mechanical coupling caused by vertical variation of formation properties. The source-independent formulation described by Eq. A-1 summarizes the conceptual basis for a robust and efficient algorithm that can be used as the forward computational model for full-waveform inversion. The transfer function T 12 depends on both the formation and borehole properties. Specifically, factors that affect the transfer function include: (a) elastic properties of the formation, (b) borehole radius R, (c) acoustic velocity V f, (d) density f, and (e) quality factors for formation and borehole fluid. Toksöz et al. (1985), Cheng et al. (1986), and Cheng (1989) have quantified some of these factors using forward-modeling techniques. Because the primary goal of this paper is to simultaneously invert radial distributions of density and P- and S-wave velocities, quality factors will not be subject to quantitative consideration. The normalized wavefield (or the normalized spectrum) of full waveforms constitutes the measurements entered into the full-waveform inversion algorithm. A cost function that enforces the quadratic difference between the simulated and measured normalized wavefields is used for inversion. To appraise the robustness of the inversion algorithm, this paper also considers a sensitivity study that quantifies the influence of noise, normalization trace, and regularization parameters. For the estimation of the radial distribution of elastic properties, density and P- and S-wave velocities are assigned a constant value within each radial layer in the near-wellbore region. Let m be the size-n vector of unknown parameters that fully describes the radial distribution of elastic properties, and let m R be a reference vector of the same size as m that has been determined from some a priori information. The estimation (inversion) of m is undertaken by minimizing a quadratic cost function, C(m), defined as (Torres- Verdín and Habashy 1994) 2C m = W d d m d obs W m m m R 2,...(A-2) where d obs is a size-m vector that contains the noisy measurements, and W d W d T is the inverse of the data covariance matrix. This data-weighting matrix describes the estimated variance for each particular measurement and the estimated correlation between measurements. The parameter denotes the prescribed value of the enforced quadratic data misfit. A priori estimates of the noise in the measurements are used to determine the magnitude of. In Eq. A-2, d(m) is the measurement vector numerically simulated for specific values of m; W m W m T is the inverse of the model covariance matrix, used to enforce both a quantitative degree of confidence in the reference model, m R, and a priori information about m; and is a Lagrange multiplier or regularization parameter. The first additive term on the right side of Eq. A-2 drives the inversion toward fitting the measurements within the desired 2 value. The sole presence of such a term in the cost function C(m) will yield multivalued solutions of the inverse problem as a result of both noisy measurements and insufficient and imperfect data sampling. Enforcing an extremely small data misfit may result in estimated models with exceedingly large norms (Torres-Verdín and Habashy 1994). The second additive term on the right side of Eq. A-2 is used to reduce nonuniqueness and to stabilize the inversion in the presence of noisy and sparse measurements. In this context, the Lagrange multiplier,, controls the relative weight of the two additive terms in the cost function. The developments considered in this paper make use of a relatively large regularization parameter at the outset, which monotonically decreases according to the number of iterations and the observed reduction of the cost function from iteration to iteration. Measurement and Model Vectors. In the cost function (Eq. A-2), the measurement vector d obs is constructed from the real and imaginary parts of the normalized spectra, Re(S) and Im(S), in the following organized fashion: d obs = Re S 1,Im S 1,...Re S j,im S j,...re S M,Im S M T,...(A-3) where j 2, 3,... M. In Eq. A-3, M is the number of actual frequency domain measurements, and the superscript T indicates transpose. The amplitude of the spectra also can be used as the measurement vector at the expense of losing phase information. The ordering procedure that assigns an index, j, to a given measurement is a function of frequency for the spectrum of each trace and receiver location. Real and imaginary parts of one measurement are arranged next to each other. If the sonic tool consists of N REC receivers, the normalized wavefield has N REC 1 traces because one trace is used to eliminate the source effect. If the number of frequencies used for each trace is N FREQ, the actual number of measurements used for the inversion is 2M 2*N FREQ *(N REC 1). Similarly, the model vector is assembled as m = V p,1, V p,2,...,v p,n, V s,1, V s,2,...,v s,n, 1,..., N T...(A-4) for the simultaneous inversion of P- and S-wave velocities and density for each radial layer, where V p,v s, and denote P-wave velocity, S-wave velocity, and density, respectively, in the radial layers numbered from 1 to N. By denoting the model parameters as m i, where i 1,2,...,3N, Eq. A-4 can be written as m = m 1, m 2,...,m 3N T....(A-5) Gauss-Newton Fixed-Point Iteration Search. A Gauss-Newton fixed-point iteration search (Gill et al. 1981) is used to determine a stationary point, m, at which the cost function attains a minimum. This method considers only first-order variations of the cost function in the neighborhood of a local iteration point. The corresponding iterated formula can be written as m k+1 = J T m k W T d W d J m k + W T m W m 1 J T m k W d T W d d m k d obs + J m k m k + W T m W m m R...(A-6) subject to l i m k+1 i u i, where i = 1,2,...N....(A-7) In these expressions, the superscript k is used as an iteration count, and J(m) is the Jacobian matrix of C(m). Upper and lower bounds enforced on m k+1 are intended to have the iterated solution yield only physically consistent results. The fixed-point iteration search for a minimum of C(m) is concluded when the measured data have been fit within the prescribed tolerance, 2. Relative data misfits computed with the formula W d d m k d obs 2...(A-8) W d d obs 2 are used in this paper to enforce a convergence criterion for the inversion. SI Metric Conversion Factors cp 1.0* E 03 Pa s cycles/sec 1.0* E+00 Hz ft 3.048* E 01 m ft E 02 m 3 in. 2.54* E+00 cm in E+01 cm 3 psi E+00 kpa *Conversion factor is exact. October 2006 SPE Reservoir Evaluation & Engineering 563