Inversion-based detection of bed boundaries for petrophysical evaluation with well logs: Applications to carbonate and organic-shale formations

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1 t Technical paper Downloaded 07/09/14 to Redistribution subject to SEG license or copyright; see Terms of Use at Inversion-based detection of bed boundaries for petrophysical evaluation with well logs: Applications to carbonate and organic-shale formations Zoya Heidari 1 and Carlos Torres-Verdín 2 Abstract Petrophysical interpretation of well logs acquired in organic shales and carbonates is challenging because of the presence of thin beds and spatially complex lithology; conventional interpretation techniques often fail in such cases. Recently introduced methods for thin-bed interpretation enable corrections for shoulder-bed effects on well logs but remain sensitive to incorrectly picked bed boundaries. We introduce a new inversion-based method to detect bed boundaries and to estimate petrophysical and compositional properties of multilayer formations from conventional well logs in the presence of thin beds, complex lithology/fluids, and kerogen. Bed boundaries and bed properties are updated in two serial inversion loops. Numerical simulation of well logs within both inversion loops explicitly takes into account differences in the volume of investigation of all well logs involved in the estimation, thereby enabling corrections for shoulder-bed effects. The successful application of the new interpretation method is documented with synthetic cases and field data acquired in thinly bedded carbonates and in the Haynesville shale-gas formation. Estimates of petrophysical/compositional properties obtained with the new interpretation method were compared to those obtained with (1) nonlinear inversion of well logs with inaccurate bed boundaries, (2) depth-by-depth inversion of well logs, and (3) core/x-ray diffraction measurements. Results indicated that the new method improves the estimation of porosity of thin beds by more than 200% in the carbonate field example and by more than 40% in the shale-gas example, compared to depth-by-depth interpretation results obtained with commercial software. This improvement in the assessment of petrophysical/compositional properties reduces uncertainty in hydrocarbon reserves and aids in the selection of hydraulic fracture locations in organic shale. Introduction Petrophysical and compositional evaluation of organic-shale and carbonate formations remains an outstanding challenge in the petroleum industry. Common well-log interpretation problems arising in organicshale and carbonate formations include presence of thin beds, extreme vertical and radial heterogeneity, and uncertainty in physical and pore-structure models. The interpretation method introduced in this paper improves conventional well-log analysis in organic-shale and carbonate formations by simultaneously correcting shoulder-bed effects and quantifying the nonlinear impact of complex lithology on well logs. Shoulder beds can significantly affect estimates of petrophysical and compositional properties in thinly bedded formations. These effects depend on factors such as bed thickness, contrast in physical properties of adjacent beds, vertical resolution of well logs included in the interpretation, and specific petrophysical and compositional properties. Experience shows that shoulder-bed effects can cause significant errors in estimates of porosity, mineral/fluid concentrations, and permeability in beds thinner than m (2 with conventional depth-by-depth well-log interpretation. This error increases with decreasing bed thickness and increasing rock heterogeneity. Conventional techniques do not effectively correct shoulder-bed effects on well-log interpretation because of the lack of fast procedures for the numerical simulation of nuclear logs. New interpretation techniques have recently been developed that explicitly correct shoulder-bed effects on low-resolution well logs and, consequently, on estimates of petrophysical and compositional formation properties (Liu et al., 2007; Sánchez-Ramirez, 2010; Heidari et al., 2012) using numerical simulation of nuclear (Mendoza et al., 2010) and electrical resistivity logs. These methods numerically simulate conventional well logs (e.g., density, neutron porosity, apparent electrical resistivity, gamma ray [GR], and photoelectric factor [PEF] logs) in 1 Texas A&M University, Harold Vance Department of Petroleum Engineering, College Station, Texas, USA. Zoya@pe.tamu.edu. 2 The University of Texas at Austin, Department of Petroleum and Geosystems Engineering, Austin, Texas, USA. cverdin@mail.utexas.edu. Manuscript received by the Editor 26 October 2013; revised manuscript received 24 January 2014; published online 8 May This paper appears in Interpretation, Vol. 2, No. 3 (August 2014); p. T129 T142, 9 FIGS., 5 TABLES Society of Exploration Geophysicists and American Association of Petroleum Geologists. All rights reserved. Interpretation / August 2014 T129

2 multilayer formations and iteratively reduce the difference between simulated and measured logs with nonlinear joint inversion algorithms. However, experience shows that inversion results could be deleteriously affected by incorrect detection of bed boundaries in thinly bedded formations. Detection of bed boundaries is often possible using high-resolution well logs (e.g., wellbore image logs) or core images. Inflection points in conventional low-resolution logs are also commonly used to detect bed boundaries. The reliability of conventional well logs for bed-boundary detection, however, is affected by shoulder-bed effects across thinly bedded formations, where low-resolution logs measure an average physical property of adjacent beds (Heidari et al., 2012). The presence of noise in well logs also affects the accuracy of bed-boundary detection methods. In such cases, quality-control indicators (and pertinent well logs, such as caliper, cable tension, delta density, etc.) could be included in the evaluation as data weights. However, in the presence of thin beds, quality-control well logs may not be adequate because they might also be affected by shoulder-bed effects. Furthermore, the choice of well log (or logs) for detection of bed boundaries is important because of differences in the corresponding volume of investigation and because of incorrect depth matching of measurements. For instance, PEF and density logs are usually preferred for bed-boundary detection due to their high vertical resolution. However, in complex lithology cases with thin beds, these two logs may not be adequate for detecting bed boundaries. Because of the cumulative effect of petrophysical and compositional properties on well logs, two adjacent beds might not exhibit a sufficiently large contrast to be distinguished from each other based on only one well log (e.g., density or electrical resistivity). Shoulder-bed effects on well logs acquired across thin adjacent beds can decrease that physical contrast even more. The combined interpretation of well logs can improve the accuracy of bed-boundary detection compared to that of conventional techniques by (1) taking into account the effects of petrophysical parameters and solid/fluid composition on different rock properties and (2) assimilating different volumes of investigation inherent to different logging tools. We introduce a double-loop algorithm for joint inversion of well logs to (1) reliably estimate bed boundaries, which enables accurate correction of shoulder-bed effects, (2) take into account the nonlinear effect of complex mineral and fluids on well logs, and (3) estimate petrophysical and compositional properties of multilayer formations in the presence of thin beds, gas, complex lithology, and kerogen (in the case of organic shale). In the following sections, we describe the proposed method and document its application in thinly bedded formations with complex lithology, including one synthetic case and two field examples. The first field example considers a hydrocarbon-bearing carbonate formation, and the second example addresses the interpretation of well logs acquired in the Haynesville shale-gas formation. Method Simultaneous assessment of bed-boundary locations and petrophysical/compositional properties We invoke two serial inversion loops for simultaneous assessment of boundaries and bed-by-bed petrophysical and compositional properties of multilayer formations. Figure 1 is a flowchart describing our proposed double-loop algorithm. The first internal loop estimates bed boundaries based on preset bed-by-bed estimates of petrophysical and compositional properties. Subsequently, the second loop takes the output of the first loop as input and updates estimates of petrophysical and compositional properties. In the next iteration, we use the updated estimates as preset values for the first internal loop. It is possible to switch the order of the two inversion loops based on the complexity of the problem. In the case of high uncertainty in estimates of bed boundaries, we first apply inversion loop no. 1 to estimate bed boundaries. Next, inversion loop no. 2 estimates petrophysical and compositional properties. However, whenever reliable a priori estimates of bed-boundaries are available (e.g., in the presence of high-resolution wellbore image logs), we recommend switching the order of the two internal loops to expedite convergence. Three approaches are suggested to initialize the bedboundary locations: (1) an initial guess based on inflection points of density or PEF logs, (2) an initial guess based on manual detection of bed boundaries using all the available well logs, and (3) an initial guess based on image logs. The number of bed boundaries is an input to the inversion. However, in the case of additional assumed bed boundaries, the thicknesses of extra beds converge to zero. In other words, false bed boundaries chosen in the initial guess will tend to be eliminated in the iteration process. The initial guess for petrophysical and compositional properties is based on interpretation procedures advanced by Heidari et al. (2012). They recommend three choices for the initial guess of petrophysical and compositional properties: (1) an initial guess based on x-ray diffraction (XRD)/core data, (2) an initial guess based on depth-by-depth nonlinear joint inversion of conventional logs (Heidari et al., 2012), or (3) an initial guess based on conventional well-log interpretation and linear/quasi-linear multimineral solvers. Petrophysical and compositional rock model In the petrophysical and compositional models assumed in this paper, the rock includes clay minerals, nonclay minerals, fluids, and kerogen (in the case of organic shale). Figure 2 shows the petrophysical and compositional rock models assumed for carbonate and organic-shale examples. We implement the dual-water T130 Interpretation / August 2014

3 Downloaded 07/09/14 to Redistribution subject to SEG license or copyright; see Terms of Use at model (Clavier et al., 1977) for resistivity-porosity-saturation calculations in the synthetic and organic shale field example and Archie s model for the carbonate field example. parameter, and x is the vector of bed-boundary locations, given by Joint inversion of well logs to detect bed boundaries This step estimates bed boundaries assuming that petrophysical and compositional properties are known beforehand. We optimize bed-boundary locations by minimizing the quadratic cost function: where nb is the number of beds and the superscript T indicates transposition. The vector of numerically simulated logs is given by CðxÞ ¼ kw d ½dðxÞ dm k22 þ α2 kxk22 ; (1) where W d is a data weighting matrix, dðxþ is the vector of numerically simulated logs, dm is the vector of available well logs, α is a regularization (stabilization) x ¼ ½x1 ; x2 ; :::; xnb 1 T ; d ¼ ½ϕN ; ρb ; PEF; GR; σ; U; Th; K T ; (2) (3) where vectors ϕn, ρb, PEF, GR, σ, U, Th, and K include nsp (number of sampling points in each well log) measurement points for neutron porosity, density, PEF, GR, apparent electrical conductivity, U (uranium), Th (thorium), and K (potassium) logs; σ is a vector that includes all the available apparent electrical conductivity Figure 1. Workflow of the interpretation method introduced in this paper. It consists of two serial inversion algorithms. The nonlinear inversion in each loop progressively improves the agreement between well logs and their numerical simulations. Inputs to the method are well logs, and outputs are bed-boundary locations, petrophysical properties, and volumetric/weight concentrations of rock mineral constituents. a) b) Figure 2. Petrophysical/compositional rock models assumed in this paper for evaluation of (a) carbonate formations and (b) organic-shale formations. Interpretation / August 2014 T131

4 logs (i.e., inverse of apparent electrical resistivity logs) with variable radial lengths of investigation. In the absence of mud-filtrate invasion (when all the electrical conductivity logs overlap), only one electrical conductivity log is input to the inversion. The size of vector d is equal to n l n sp, where n l is the number of well logs included in the inversion. In the implementation described above, the data-weighting matrix controls the impact of different well logs on inversion results for bed-boundary locations and is given by 2 3 W d ¼ 6 4 ½w 1 ði nsp n sp ÞŠ ; (4) 0 ½w nl ði nsp n sp ÞŠ where I is the unity matrix and w i is the weight associated with well log i. The purpose of the data-weighting matrix is to (1) normalize the various log measurement scales and (2) control the relative impact of each well log on inversion results. Accordingly, assigning a small value to w j can decrease the effect of well-log j on inversion results. We minimize the quadratic cost function, CðxÞ, using Levenberg-Marquardt s method (Marquardt, 1963). To apply this gradient-based technique, we first numerically calculate the Jacobian matrix at every linear iteration. The corresponding entries of the Jacobian matrix, J, are given by J ij ¼ d i x j ; 1 i n l n sp ; 1 j n b : (5) A stabilization parameter is selected with Hansen s (1994) L-curve strategy. At every linear iteration, bedboundary locations are updated based on the calculated Jacobian matrix together with the difference between well logs and their numerical simulations. The convergence criterion is satisfied if (1) the relative difference between the norm of data residuals yielded by two subsequent iterations is less than 0.01%, (2) the maximum difference between bed boundaries estimated in two subsequent iterations is less than 0.01 ft, or (3) after reaching a prescribed maximum number of iterations. Assessment of bed-by-bed petrophysical and compositional properties via nonlinear joint inversion of well logs Heidari et al. (2012) introduce a new method for bed-by-bed joint inversion of well logs to estimate petrophysical and compositional properties of multilayer formations. The main advantages of this method compared to conventional depth-by-depth interpretation techniques are as follows: (1) explicit assessment of the nonlinear correlation between well-log measurements and physical properties of pure formation components and (2) implicit correction of shoulder-bed effects on well logs. The method takes bed-boundary locations and conventional well logs as input. In the first step, bed-by-bed physical properties (i.e., electrical resistivity, neutron migration length, density, PEF, and U, Th, and K concentrations) are estimated using separate inversion of conventional well logs (i.e., apparent electrical resistivity, neutron porosity, density, PEF, and GR/GR-spectroscopy logs). Bed-by-bed physical properties are then used in a bed-by-bed joint inversion process to estimate petrophysical and compositional properties. The inversion begins with an initial guess for petrophysical and compositional properties. Schlumberger s commercial software, SNUPAR (mark of Schlumberger, McKeon and Scott, 1989), calculates neutron migration length and PEF based on volumetric concentrations and chemical compositions of bedby-bed formation components. We iteratively update bed-by-bed petrophysical and compositional properties to minimize the differences between estimated physical properties and their numerical simulations using a gradient-based minimization algorithm (Heidari et al., 2012). Synthetic case A synthetic case is constructed to replicate formation properties in a carbonate reservoir. The objective is to investigate the efficiency of our bed-boundary detection method in the presence of (1) shoulder-bed effects, (2) thin beds, (3) closely spaced thin beds, (4) gas, and (5) complex lithology. Mineral and fluid constituents in the formation include quartz, calcite, dolomite, chlorite, bound water, and gas. Well logs input to the inversion are array-induction apparent resistivity, neutron porosity, density, PEF, and GR spectroscopy (Th, Ur, and K logs). Table 1 summarizes the assumed formation properties; mud-filtrate invasion is assumed negligible, and the depth-sampling rate is m (0.25 for all the well logs. Bed-boundary locations Table 1. Synthetic case: Summary of assumed Archie s parameters and fluid and formation properties. Variable Value Units Winsauer factor in Archie s equation, a 1.00 ( ) Archie s porosity exponent, m 2.00 ( ) Archie s saturation exponent, n 2.00 ( ) Connate-water salt concentration 80 kppm NaCl Bound-water salt concentration 100 kppm NaCl In situ water density 1.00 g cm 3 In situ gas density 0.19 g cm 3 Formation temperature 230 F Shale porosity 0.10 ( ) Volumetric concentration of clay in shale 0.50 ( ) Wet shale density 2.64 g cm 3 Wellbore radius cm T132 Interpretation / August 2014

5 and petrophysical/compositional properties were simultaneously estimated using the above-described double-loop inversion method. Figure 3 shows the model, initial guess, and final estimates of the bed-boundary locations. The same figure compares the model (actual) to numerically simulated logs. Simulated logs are plotted in connection with initial and final estimates of bed-boundary locations and petrophysical/compositional properties. We assumed similar petrophysical and compositional properties in the top three permeable beds (Figure 4). However, numerically simulated well logs show different physical properties in these three beds due to significant shoulder-bed effects. The choice of a parsimonious (constant) initial guess for bed-boundary locations verifies the stability of the inversion method, whereas the initial guess for petrophysical and compositional properties is constructed with results obtained from depth-by-depth nonlinear joint inversion of well logs (Heidari et al., 2012). We then use center-bed values (based on the initial guess for bed-boundary locations) as the initial guess for petrophysical and compositional properties. Figure 4 compares the actual petrophysical/compositional model together with the corresponding estimates of porosity, water saturation, volumetric concentration of shale, and volumetric concentrations of quartz, calcite, and dolomite obtained from (1) depth-by-depth nonlinear joint inversion of well logs, (2) the introduced double-loop serial joint inversion of well logs, and (3) bed-by-bed nonlinear joint inversion of well logs with inaccurate bed boundaries. The double-loop serial joint inversion of well logs successfully converged to the actual petrophysical and compositional properties. Results confirm that the inversion method implicitly corrects shoulder-bed effects on well logs by taking into account the geometrical configuration of logging tools in the numerical simulation of well logs. Without a reliable bed-boundary detection technique, it is possible to overlook the bed boundary located at m (17 ft, relative depth), which gives rise to approximately 12% and 24% relative errors in estimates of nonshale porosity and nonshale water saturation using the bed-by-bed inversion method, respectively. We also observe a significant underestimate of porosity in the m (0.5 bed located at m (12 ft, relative depth) using depthby-depth and bed-by-bed interpretation methods when bed-boundary locations are inaccurate. a) b) c) d) e) Figure 3. Synthetic case: Comparison of final numerically simulated well logs (dashed-dotted black line), input well logs (solid line), and numerically simulated well logs for the initial guess (dashed line). Results are shown for array-induction resistivity (b), PEF (c), GR (d), density and neutron porosity (water-filled limestone porosity units, and (e) logs. Panel (a) shows assumed values (black solid line), the initial guess (dashed green line), and final estimates (dashed-dotted red line) of the bedboundary locations. Interpretation / August 2014 T133

6 Downloaded 07/09/14 to Redistribution subject to SEG license or copyright; see Terms of Use at Tables 2 and 3 list (1) actual values, (2) initial guess, (3) final estimates of bed-boundary locations, nonshale porosity, nonshale water saturation, volumetric concentration of shale, and volumetric concentrations of quartz, calcite, and dolomite, and (4) the absolute error on final estimates of petrophysical and compositional properties. Convergence is achieved after five iterations of the main loop (including the two internal inversion loops). The maximum error in the final estimates of bed-boundary locations, nonshale porosity, Figure 4. Synthetic case: Comparison of the final estimates of porosity, fluid saturations, and volumetric concentrations of minerals assumed in the actual model (b) and those obtained from depth-by-depth nonlinear joint inversion of well logs (c), the introduced nonlinear bed-by-bed nonlinear joint inversion of well logs after accurate assessment of bed-boundary locations (d), and bed-by-bed nonlinear joint inversion of well logs with inaccurate bed-boundary locations (e). Panel (a) shows bed-boundary locations assumed in the model (solid black lines), final estimates of bed-boundary locations using the proposed method (dashdotted red lines), and perturbed bed-boundary locations (green dashed lines). Table 2. Synthetic case: Comparison of actual values, initial guess, and final estimates of bed-boundary locations after simultaneous assessment of bed-boundary locations and petrophysical and compositional properties of the synthetic multilayer formation. The last row lists the uncertainty of final estimates of bed-boundary locations corresponding to 5% zero-mean Gaussian random perturbations on the original synthetic well logs. Relative bed-boundary locations and their uncertainty (m) Actual location (m) (3.000 (5.000 (8.000 (9.000 ( ( ( ( ( ( T134 Interpretation / August 2014 Initial guess (m) (2.000 (6.000 (7.000 ( ( ( ( ( ( ( Final estimates (m) (3.000 (4.994 (7.990 (8.999 ( ( ( ( ( ( Uncertainty (m) ( ( ( ( ( ( ( ( ( ( 0.021

7 and nonshale water saturation is lower than m (0.01 ft, absolute error), 0.1 porosity units (absolute error), and 0.1 saturation units (absolute error), respectively. Figure 5 describes the uncertainty in estimates of bed-boundary locations corresponding to 5% zero-mean Gaussian random perturbations on the original synthetic well logs. A maximum uncertainty of approximately 4.57 cm (1.8 in) arises in the subsequent thin beds located in the depth interval of m ( We also investigated the sensitivity of the bed-boundary detection loop (inversion loop no. 1) to estimates of bed-by-bed petrophysical and compositional properties by perturbing actual properties with 5% additive zero-mean Gaussian random variations of their original value. Results from this exercise indicate high stability of the introduced algorithm for bed-boundary detection. However, experience shows that a decrease in bed thickness increases the uncertainty of bed-boundary detection. Specifically, 15% relative uncertainty in the assessment of bed-by-bed petrophysical and compositional properties gives rise to a maximum error of m (0.35 in the corresponding estimates of bed-boundary locations for a m (1 bed surrounded by subsequent beds. Field example no. 1: Hydrocarbonbearing carbonate formation Field example no. 1 is intended to verify the reliability of the introduced method in the petrophysical/ compositional evaluation of a challenging carbonate formation. We select a thinly bedded depth interval in this oil-bearing carbonate formation where conventional well-log interpretation methods are not reliable. The well was drilled with oil-based mud (OBM), and the well-log sampling rate is m (0.5. We assume that the effect of the mud-filtrate invasion on well logs is negligible due to overlapped array-induction apparent resistivity logs. Based on core/xrd measurements, the assumed components of the formation consist of (1) nonclay minerals, specifically, quartz, calcite, and dolomite, (2) clay minerals (smectite and chlorite) in Table 3. Synthetic case: Comparison of actual values, initial guess, and final estimates of nonshale porosity, nonshale water saturation, volumetric concentration of shale, and volumetric concentrations of quartz, calcite, and dolomite after simultaneous assessment of bed-boundary locations and petrophysical and compositional properties of the multilayer formation. Permeable beds Bed number Nonshale porosity Actual value Initial guess Final estimates Absolute error Nonshale water saturation Actual value Initial guess Final estimates Absolute error Volumetric concentration of shale Actual value Initial guess Final estimates Absolute error Volumetric concentration of quartz Actual value Initial guess Final estimates Absolute error Volumetric concentration of calcite Actual value Initial guess Final estimates Absolute error Volumetric concentration of dolomite Actual value Initial guess Final estimates Absolute error Interpretation / August 2014 T135

8 the thinly bedded depth interval, and (3) gas and saline water. Inputs to the joint inversion technique consist of (1) well logs including array-induction resistivity, PEF, density, and neutron porosity and (2) assumed formation properties such as Archie s parameters and matrix, fluid, and formation properties, listed in Table 4. We assume that Archie s equation is reliable in the depth interval of interest in this carbonate formation due to the negligible impact of isolated pore space and fractures on electrical resistivity measurements. The constants in Archie s equation were estimated from core measurements. However, more information about pore texture from core evaluation can improve the resistivity-porosity-saturation model. Figure 6 shows the final estimates of bed-boundary locations as well as the measured and numerically simulated well logs after two iterations of the main loop. We estimated compressional-wave slowness based on the final estimates of petrophysical and compositional properties. The agreement between estimated and measured compressional-wave slowness serves to verify the estimations of porosity, fluid saturations, and volumetric concentrations of minerals. In addition, the comparison indicates nonnegligible shoulder-bed effects on the compressional wave slowness log. Figure 7 compares the final estimates of total porosity, total water saturation, and volumetric concentrations of quartz, calcite, dolomite, and clay against core/xrd measurements and corresponding depth-by-depth estimates obtained with commercial software. The new method improves porosity estimates by more than 200% in the thin bed located at m (10 ft, relative depth), while the depth-by-depth well-log interpretation is not reliable due to significant shoulder-bed effects. A perturbation of m (0.25 foot) in the bed-boundary location of this thin bed causes a 447% relative error in porosity estimates, when using the bed-by-bed inversion method. Table 4. Field example no. 1: Summary of assumed Archie s parameters and matrix, fluid, and formation properties. Variable Value Units Winsauer factor in Archie s equation, a 1.00 ( ) Archie s porosity exponent, m 2.50 ( ) Archie s saturation exponent, n 3.00 ( ) Connate-water salt concentration 230 kppm NaCl In situ water density 1.00 g cm 3 In situ oil density 0.98 g cm 3 Wet clay density 2.81 g cm 3 Formation temperature 320 F Wellbore radius 15.5 cm Figure 5. Synthetic case: Uncertainty of estimates of bed-boundary locations due to 5% zero-mean Gaussian random perturbations on the original synthetic well logs. T136 Interpretation / August 2014

9 Field example no. 2: Haynesville shale-gas formation An important application of the interpretation method advanced in this paper is in the assessment of organic-shale formations because of the common presence of thin beds and spatially varying lithology. Moreover, this field example is intended to examine the reliability of the introduced method for bedboundary detection and petrophysical/compositional evaluation in organic-shale formations. The Haynesville formation is a late-jurassic formation with total organic carbon (TOC) between 2% and 5% (weight concentration) and clay volumetric concentration of lower than 50% (Quirein et al., 2010; Spain et al., 2010). Table 5 summarizes the assumed Archie s parameters and matrix, fluid, and formation properties in field example no. 2. We expect 0.1 g cm 3 uncertainty in fluid densities. However, we found that the impact of this uncertainty was negligible on final inversion results. The well was drilled with water-based mud (WBM), but we assume a negligible impact of the process of mud-filtrate invasion on well logs. Input logs include array-induction resistivity, density, neutron porosity, and PEF. We do not include the GR log in the inversion due to uncertainty in U, Th, and K concentrations of kerogen and mineral constituents in the formation. The assumed components consist of (1) nonclay minerals, namely, quartz, calcite, plagioclase, pyrite, and negligible amount of dolomite, (2) clay minerals, specifically, illite and chlorite, (3) kerogen type II, and (4) gas and saline water. Outputs from the inversion are weight concentrations of rock constituents. Because of the large number of unknown parameters compared to the number of available well logs, this example constitutes an underdetermined inverse problem. To mitigate the nonuniqueness of the inversion results in this underdetermined estimation problem, we suggest (1) choosing the initial guess for bed-boundary locations based on inflection points in the PEF log, (2) initializing the inversion with estimates of bed-by-bed petrophysical and compositional properties obtained from the initial guess of bed-boundary locations, and (3) implementing constraints on the volumetric concentrations of mineral constituents. As suggested by Heidari and Torres-Verdín (2013), XRD data in this formation indicate a linear relationship between weight concentrations of quartz and plagioclase given by a) b) c) d) e) f) W plag: ¼ 0.32 W quartz ; (6) Figure 6. Field example no. 1: Comparison of final numerically simulated (dashed-dotted black line) and available well logs (solid line). Results are shown for array-induction deep apparent resistivity (d), PEF (e), and density and neutron porosity (water-filled limestone porosity units, [f]) logs. Panels (a and b) show estimates of bed-boundary locations and measured GR, respectively. Panel (c) shows the measured and estimated bed-by-bed compressional-wave slowness (DTCO). Interpretation / August 2014 T137

10 and a linear relationship between weight concentrations of illite and chlorite, given by W illite ¼ 2.1 W chlorite ; (7) where W plag: is the weight concentration of plagioclase, W quartz is the weight concentration of quartz, W illite is the weight concentration of illite, and W chlorite is the weight concentration of chlorite. Constraints on the volumetric concentrations of mineral constituents can be applied based on core XRD measurements. For instance, the volumetric concentration of pyrite is assumed to be less than 10% in this case. Figure 8 shows the final estimates of bed-boundary locations together with available well logs and their corresponding numerical simulations. Figure 9 compares the final estimates of total porosity, total water saturation, and solid weight concentrations of kerogen and minerals obtained with the new interpretation method, commercial software, and core/xrd measurements. Uncertainty bars on estimates of petrophysical and compositional properties originate from 5% zero-mean Gaussian random perturbations of the measured well logs. The new method improves the estimates of porosity and water saturation as well as the corresponding volumetric concentrations of minerals with respect to those of commercial software. Furthermore, the commercial software requires significant calibration efforts using core measurements. In the new method, we invoke core measurements only to estimate constant parameters in the resistivity-saturation-porosity models and to develop constraints to mitigate nonuniqueness of results. Calibration efforts with commercial software can fail when few core measurements are available Table 5. Field example no. 2: Summary of assumed Archie s parameters and matrix, fluid, and formation properties. Variable Value Units Winsauer factor in Archie s equation, a 1.00 ( ) Archie s porosity exponent, m 1.60 ( ) Archie s saturation exponent, n 2.00 ( ) Connate-water salt concentration 200 kppm NaCl Bound-water salt concentration 200 kppm NaCl In situ water density 1.00 g cm 3 In situ gas density 0.19 g cm 3 Kerogen density 1.2 g cm 3 Dry clay density 2.84 g cm 3 Formation temperature 265 F Wellbore radius cm a) b) c) d) e) f) Figure 7. Field example no. 1: Comparison of estimated petrophysical and compositional properties obtained with the doubleloop nonlinear inversion method introduced in this paper (solid red lines), depth-by-depth inversion obtained with commercial software (dashed blue line) and core/xrd data (blue dots). Results are shown for total porosity (a); total water saturation (b); and volumetric concentrations of quartz (c), calcite (d), dolomite (e), and clay (f). T138 Interpretation / August 2014

11 and rocks exhibit rapid variations of petrophysical and compositional properties. Discussion In a previous publication, we introduced a new method for bed-by-bed assessment of petrophysical and compositional properties of the formation (Heidari et al., 2012). However, results obtained using our new technique were highly sensitive to bed-boundary locations. In this paper, we introduced an inversion-based method that simultaneously estimates bed-boundary locations and bed-by-bed petrophysical/compositional properties of the formation. Inputs to inversion include conventional well logs, the initial guess for bed-boundary locations and bed-by-bed petrophysical/compositional properties, and petrophysical models designed for different formations. An advantage of the introduced method over conventional bed-boundary detection techniques is the possibility of using all the available high-quality well logs when detecting bed boundaries, instead of using only one well log (the one previously qualified as the most sensitive to bed boundaries). This advantage makes the technique generally applicable in a wide variety of challenging formations. Furthermore, it enables taking into account contrasts between different physical rock properties to detect bed boundaries. We described two successful field applications of the introduced method in carbonate and shale-gas formations exhibiting thin beds and complex lithology. The performance of the introduced method was evaluated by comparing the estimated petrophysical and compositional properties against core measurements. Even though actual locations of the bed boundaries were not available for the comparison against estimated locations, a good agreement between the estimated petrophysical/compositional properties of the formation can be indicative of reliable bed-boundary locations. However, there is uncertainty on the number of bed boundaries, which can impact the performance of the introduced technique, if the initial guess for the number of petrophysical beds is less than the actual number. Synthetic and field examples indicated a reliable performance of the introduced method in challenging reservoirs including thin beds and complex lithology. However, similar to other conventional methods for assessment of bed-boundary locations and petrophysical properties, there are limitations associated with this method in the presence of (1) uncertainty in logging depth, (2) unreliable well-log measurements caused by environmental effects, (3) nonuniqueness of the results, and (4) uncertainty in petrophysical models (e.g., resistivity-porosity-saturation models in carbonate and organic-shale formations). It was also found that the a) b) c) d) e) Figure 8. Field example no. 2: Comparison of final numerically simulated (dashed-dotted black line) and available well logs (solid line). Results are shown for array-induction deep resistivity (c), PEF (d), and density and neutron porosity (water-filled limestone porosity units, [e]) logs. Panels (a and b) show estimates of bed-boundary locations and measured GR, respectively. Interpretation / August 2014 T139

12 new interpretation method will not provide reliable bed-boundary locations and petrophysical properties (1) when beds are thinner than two times the depthsampling interval or (2) in the absence of a measurable property contrast between adjacent beds. Nonuniqueness of the results is a typical challenge in well-log-based petrophysical evaluation of formations with complex lithology, where the number of unknown parameters is greater than the number of well logs. Applying constraints based on core measurements can mitigate nonuniqueness of inversion results. Furthermore, a realistic initial guess for bed-boundary locations and petrophysical/compositional properties can avoid trapping into local minima. In the presence of nonuniqueness, comparison of final results to core measurements and image logs (for bed-boundary locations) can improve the reliability of final inversion results. It is also possible to perform the inversion with different initial guesses and examine the variability of inversion results as an indication of nonuniqueness. In both field examples, we assumed that Archie s and dual-water equations were reliable to correlate rock electrical resistivity to water saturation in the depth interval of interest. This assumption was validated in the case of shale-gas formation by comparing the estimated water saturation against core measurements, but was not validated in the carbonate formation due to lack of core measurements. However, the sensitivity of the estimated lithology and porosity to hydrocarbon/ water saturation is negligible in the carbonate field example because of hydrocarbon density (i.e., approximately 0.9 g cm 3 ) close to water density in this case. Thus, the assumption of Archie s equation for the carbonate example does not impact the estimates of porosity, lithology, and bed-boundary locations. The resistivity-porosity-saturation model can, however, be substituted by any other reliable resistivity-water saturation-porosity model. The impact of mud-filtrate invasion on well logs was negligible; hence, it is not taken into account in this paper. However, it is possible to assimilate the effect of invasion in the proposed method by numerically simulating the process of invasion in the near-wellbore region (Heidari and Torres-Verdín, 2013) at the expense of much larger computer time. Conclusions We documented the successful application of a new method for simultaneous bed-boundary detection and petrophysical evaluation of thinly bedded organic-shale and carbonate formations. Results confirmed that bedby-bed well-log interpretation significantly improves the evaluation of thinly bedded formations compared to conventional depth-by-depth interpretation methods. a) b) c) d) e) f) g) h) i) Figure 9. Comparison of estimates of petrophysical and compositional properties obtained with the bed-by-bed nonlinear inversion method introduced in this paper (solid red lines), depth-by-depth inversion using commercial software (dashed blue line), and core/xrd data (blue dots). Uncertainty bars on estimates of petrophysical and compositional properties are due to 5% zero-mean Gaussian random perturbations on the original measured well logs. Results are shown for total porosity (a); total water saturation (b); and solid weight concentrations of quartz (c), calcite (d), kerogen (e), plagioclase (f), illite (g), chlorite (h), and pyrite (i). T140 Interpretation / August 2014

13 However, it was found that an uncertainty of m (0.25 in the detection of boundaries for a m (0.5 bed can cause 35% and 48% relative errors in estimates of nonshale porosity and nonshale water saturation, respectively. A synthetic interpretation example confirmed the stability and accuracy of the new method for assessment of bed-boundary locations and bed-by-bed formation properties across thin beds. The reliability of the method was verified for beds with a thickness greater than two times the depth-sampling interval. The uncertainty of the inversion results will significantly increase in beds thinner than two times the depth-sampling interval. This limitation also exists in conventional methods such as the inflection-point technique. We applied the introduced method to two field applications of carbonate and shale-gas formations. Estimates of porosity improved by more than 200% (relative improvement) and 30% compared to depth-by-depth interpretation techniques in the carbonate and the shale-gas formations, respectively. The synthetic and field examples studied in this paper verified that reliable assessment of bed boundaries combined with bed-by-bed interpretation of low-resolution well logs improves the estimates of bed-by-bed petrophysical and compositional properties, thereby reducing the uncertainty in the assessment of hydrocarbon reserves. Acknowledgments The work reported in this paper was funded by The University of Texas at Austin s Research Consortium on Formation Evaluation, jointly sponsored by Anadarko, Apache, Aramco, Baker-Hughes, BG, BHP Billiton, BP, Chevron, ConocoPhillips, ENI, ExxonMobil, Halliburton, Hess, Maersk, Marathon Oil Corporation, Mexican Institute for Petroleum, Nexen, ONGC, Petrobras, Repsol, RWE, Schlumberger, Shell, Statoil, Total, and Weatherford. Special thanks go to BP for providing some of the field data reported in this paper. List of symbols a = Winsauer factor in Archie s equation ( ) CðxÞ = Cost function ( ) d = Vector of simulated logs d m = Vector of measured or model logs I = Unity matrix ( ) J = Jacobian matrix m = Archie s porosity exponent ( ) n = Archie s saturation exponent ( ) n b = Number of beds ( ) n l = Number of well logs ( ) n sp = Number of sampling points in each well log ( ) R = Apparent resistivity measurements (ohm-m) S w = Total water saturation ( ) V r = Volume of rock ( ) V sh = Volumetric concentration of shale ( ) W chlorite = Weight concentration of chlorite ( ) W d = Data weighting matrix ( ) W illite = Weight concentration of illite ( ) W plag: = Weight concentration of plagioclase ( ) W quartz = Weight concentration of quartz ( ) α = Regularization parameter ( ) ϕ N = Neutron porosity (V/V) ϕ s = Nonshale porosity ( ) ρ b = Bulk density (g cm 3 ) σ = Electrical conductivity (S m) List of acronyms DTCO = Delta-T compressional GR = Gamma-ray K = Potassium kppm = Kiloparts per million OBM = Oil-base mud PEF = Photoelectric factor SNUPAR = Schlumberger Nuclear Parameter code Th = Thorium TOC = Total organic carbon U = Uranium WBM = Water-based mud XRD = X-ray diffraction References Clavier, C., G. Coates, and J. Dumanoir, 1977, The theoretical and experimental basis for the dual water model for the interpretation of shaly sands: Presented at AIME Annual Technical Conference and Exhibition: SPE, Hansen, P. C., 1994, Regularization tools: A MATLAB package for analysis and solution of discrete ill-posed problems: Numerical Algorithms, 6,1 35, doi: / BF Heidari, Z., and C. Torres-Verdín, 2013, Inversion-based method for estimating total organic carbon and porosity and for diagnosing mineral constituents from multiple well logs in shale-gas formations: Interpretation, 1, no. 1, T113 T123, doi: /INT Heidari, Z., C. Torres-Verdín, and W. E. Preeg, 2012, Improved estimation of mineral and fluid volumetric concentrations from well logs in thinly-bedded and invaded formations: Geophysics, 77, no. 3, WA79 WA98, doi: /geo Liu, Z., C. Torres-Verdín, G. L. Wang, A. Mendoza, P. Zhu, and R. Terry, 2007, Joint inversion of density and resistivity logs for the improved petrophysical assessment of thinly-bedded clastic rock formations: Presented at SPWLA 48th Annual Logging Symposium, paper VV. Marquardt, D. W., 1963, An algorithm for least-squares estimation of nonlinear parameters: SIAM Journal of Applied Mathematics, 11, McKeon, D. C., and H. D. Scott, 1989, SNUPAR A nuclear parameter code for nuclear geophysics applications: IEEE Transactions on Nuclear Science, 36, no. 1, , doi: / Interpretation / August 2014 T141

14 Mendoza, A., C. Torres-Verdín, and W. E. Preeg, 2010, Linear iterative refinement method for the rapid simulation of borehole nuclear measurements, Part I: Vertical wells: Geophysics, 75, no. 1, E9 E29, doi: / Quirein, J., J. Witkowsky, J. Truax, J. Galford, D. Spain, and T. Odumosu, 2010, Integrating core data and wireline geochemical data for formation evaluation and characterization of shale gas reservoirs: Presented at SPE Annual Technical Conference and Exhibition, Sánchez-Ramirez, J. A., C. Torres-Verdín, G. L. Wang, A. Mendoza, D. Wolf, Z. Liu, and G. Schell, 2010, Field examples of the combined petrophysical inversion of gamma-ray, density, and resistivity logs acquired in thinly-bedded clastic rock formations: Petrophysics, 51, Spain, D. R., and G. A. Anderson, 2010, Controls on reservoir quality and productivity in the Haynesville Shale, northwestern Gulf of Mexico Basin: Gulf Coast Association of Geological Societies Transactions, 60, Zoya Heidari received a Ph.D. (2011) in petroleum engineering from the University of Texas at Austin. She is an assistant professor in the Petroleum Engineering Department of Texas A&M University in College Station and the Chevron Corporation faculty fellow in petroleum engineering. She is the founder and director of the Texas A&M Joint Industry Research Program on Multi-Scale Formation Evaluation of Unconventional and Carbonate Reservoirs. Her research interests include petrophysics, well-log interpretation, borehole geophysics, inverse problems, rock physics, and reservoir characterization of unconventional reservoirs. Carlos Torres-Verdín received a Ph.D. (1991) in engineering geoscience from the University of California at Berkeley. During , he held the position of research scientist with Schlumberger-Doll Research. From 1997 to 1999, he was reservoir specialist and technology champion with YPF (Buenos Aires, Argentina). Since 1999, he has been affiliated with the Department of Petroleum and Geosystems Engineering of the University of Texas at Austin, where he is currently full professor, holds the Zarrow Centennial Professorship in Petroleum Engineering, and conducts research on borehole geophysics, formation evaluation, well logging, and integrated reservoir characterization. He is the founder and director of the Research Consortium on Formation Evaluation at the University of Texas at Austin, which is currently sponsored by 32 companies. He has published more than 130 refereed journal papers, 180 conference papers, and is author of two patents, has served as guest editor for Radio Science, as associate editor for the Journal of Electromagnetic Waves and Applications, SPE Journal, and Petrophysics (SPWLA) and is currently associate editor for GEOPHYSICS and editorial board member of The Leading Edge. He is a recipient of the 2014 Gold Medal for Technical Achievement from the SPWLA, of the 2006 Distinguished Technical Achievement Award from the SPWLA, of the 2008 Formation Evaluation Award from the SPE, of the 2003, 2004, 2006, and 2007 Best Paper Awards in Petrophysics by the SPWLA, of the 2006 Best Presentation Award and the 2007 Best Poster Award by the SPWLA. T142 Interpretation / August 2014