- 1-3D Petrophysical Modeling Usning Complex Seismic Attributes and Limited Well Log Data Mehdi Eftekhari Far, and De-Hua Han, Rock Physics Lab, University of Houston Summary A method for 3D modeling and interpretation of log properties from complex seismic attributes (obtained from 3D post stack seismic data) is developed by integrating Principal Component Analysis and Local Linear Modeling. Complex seismic attributes have non-linear relationships with petrophysical properties of rocks. These complicated relationships can be approximated using statistical methods. This method has been tested successfully on real data sets with different lithology and geology settings. Log properties (sonic, gamma ray density etc.) were predicted at the location of the second well (blind well test). It has proved to work with limited log information (data from one well) whereas conventional methods, such as geostatistics, used for this purpose need well log information from several wells to correlate seismic and well data. Once the performance of the model is verified by blind well test, 3D log volumes can be calculated from 3D seismic attribute data. Rock physics theory and available core and well log information can be used to define rock physics templates (RPTs). Using RPTs, the modeling results of the statistical method are interpreted and also uncertainty of predictions is quantified at places with no well control, in a meaningful way, by considering predictions of different log types simultaneously. Introduction Rock physics and petrophysics rules provide the framework for a reliable seismic inversion, modeling and interpretation of rock properties. Any statistical method is only valid if it is explainable in terms of the underlying physics which control rock and fluids behavior. Rock physics is a useful tool for quantitative interpretation which provides the basic relationship between lithology, fluid content, depth, pressure, stress history, geological setting of the reservoir etc. Rock physics can be utilized to build a template for efficient reservoir characterization (Ǿdegaard and Avseth, 2004; Avseth et al, 2006; Andersen and Wijngaarden, 2007). By identifying different clusters in the rock property cross-plots, we can predict the lithology of the reservoir. Seismic data are routinely and effectively used to estimate the structure of reservoir bodies. Also they have been increasingly used for estimating the spatial distribution of rock properties. Many authors have investigated the possible relations between individual seismic attributes and rock properties (or structural indications). Taner et al. 1994 published a list of such relations. The idea of using multiple seismic attributes to predict log properties was first proposed by Schultz et al. (1994). They used log data from 15 wells to train with seismic attributes and predict rock properties. Hampson et al. (2001) also explains use of different neural networks for multi-attribute analysis and reservoir property prediction. While we know that seismic signals features are directly caused by rock physics phenomena, the links between these two are complex and difficult to derive theoretically. Seismic response depends on many variables, such as temperature, volume of clay, overburden,
- 2 - pressure, nature and geometry of the layering, and other factors which affect elastic and absorption response. These complex relations can vary from one layer to another, and even within a single layer or reservoir compartment (Schultz et al., 1994). Schultz et al. (1994) showed that simultaneous use of seismic attributes with well log data leads to better prediction of reservoir or rock properties, compared to estimations using only well data. A reasonable way to combine seismic attributes and well log data for property prediction should include a statistical method. Data description Two data sets have been used in this study. Seismic data, which was used for inversion of petrophysical properties, belongs to a sand/shale reservoir from the Middle East. Due to lack of core data, another dataset from Teapot dome in Wyoming is considered for future research and testing. Preliminary research has been done on rock physics properties of cored parts (well 48-X-28 from depth of ~5300 to ~ 5650 ft) to define RPTs. There are about 900 wells drilled in Teapot dome. Current producing part is Tensleep formation which is mainly sandstone with thin sequences of siltstone, dolostone, anhydrite and limestone. Some details of the Tensleep formation is listed in the table 1. Rock Physics Templates Using available core data from well 48-X-28, rock physics templates (RPT) were defined. Preliminary study of well log information shows that clay content, porosity and compaction are important factors controlling the properties of rocks. Therefore, in order to distinguish different lithologies using templates, they should be defined in such a way to include these factors.. Shaliness (clay content) particularly has proved to be a good indicator of lithology especially as seen in figure (1), the difference between anhydrite and limestone is not distinguishable from Vp- Acoustic Impedance (AI) and NPhi-AI cross plots but when we use Gamma Ray (GR)-AI and GR-NPhi cross plots, anhydrite and limestone lithologies separate nicely (color bars in all figures represent depth in feet). Also Vp-NPhi cross plot of sandstone obtained from well logs is compared to the classical results for clean sand stone as shown in figure (2) (Han and Batzle, 2004 and Marion 1990). Some other templates are shown in figure (3). Well logs usually contain significant information about lithology and fluid properties. By crossplotting measured acoustic impedance and Vp, porosity, clay content etc. against each other and knowing the depth of measurements and lithology types from cores, reliable RPTs can be obtained by which one can differentiate lithology and fluid properties. To construct the rock physics template, the rock physics model and parameters should be carefully chosen for the different zones around the target reservoirs. Different zones will have variable temperatures, pressures, and lithologies. Theoretical modeling for defining the bounds of rock properties can be done using Hertz-Mindlin contact theory (Mindlin, 1949) to construct the dry modulus of the rock at the high porosity end number, which is pressure dependent. The other end point for modulus of the dry rock at zero porosity is selected to be the modulus of the solid mineral. Between the high porosity and zero porosity end members, the modulus of the dry rock can be estimated based on the Hashin-
- 3 - Shtrikman (1963) bounds, which give the theoretical predictions of the effective elastic modulus of a mixture of grains and pores. Figure 1: RPTs for anhydrite+limestone in Tensleep formation. a) Vp-AI, b) Vp-NPhi, c) GR-NPhi, d) GR-AI (color bars in all figures represent depth in feet). Anhydrite and limestone are distinguishable when using RPTs containing shaliness. Figure 2: a) Cross plot Vp-Porosity for clean sandstone (lab measurements). b) Cross plot of Vp-NPhi for Tensleep formation (from well logs) Figure 3: RPTs for different rock types in Tensleep formation; RPTs in first column are Vp-NPhi, second: GR-AI, and third: Vp-AI. Rock types are shown on top of each plot.
- 4 - Statistical Modeling The proposed statistical method has three main steps; 1- Seismic-Well Tying and attribute selection 2-Principal Component Analysis (PCA), 3-Extracting Statistical Relationships between Seismic Attributes and Well Logs. -Seismic-Well Tying and Attribute selection: In order to find the relation between attributes and log properties, the first step is to tie well information to seismic data by check-shot correction and correlating synthetic with seismic traces. Even if we do an excellent job in wavelet extraction and building synthetic traces, there are still some unwanted events (multiples) in seismic which will mislead our modeling. VSP data (stacked trace) can be used to avoid this problem. Selection of an optimum set of seismic attributes which can lead to best prediction of rock properties from seismic data has been a major challenge in multi-attribute analysis. Step-wise regression (Draper, and Smith, 1966) is widely used for this purpose but it is not the optimal way for doing that. It is based on the correlation analysis of attributes and it usually leads to a local maximum (rather than global maximum). Based on the experiments on several data sets, an iterative algorithm is proposed for attribute selection. It was concluded that the choice of clustering method and also the number of dimensions which are ignored after PCA are as important as the selection of attributes. We start with seismic attributes which are believed to be more correlated to rock properties (based on correlation analysis of each attribute with well log information), and then different combinations are tried for modeling in an iterative manner and modeling result is monitored until the optimum set and also optimum number of dimensions to be ignored, are determined. Some of the seismic attribute used for this study which gave the best modeling results are (Taner et al., 1979 and Taner et al. 1994): o Amplitude weighted average frequency o Quadrature trace o Instantaneous Frequency o First and second derivative of amplitude o Instantaneous phase o Dominant frequency -PCA: Using a set of seismic attributes (instead of one attribute) for rock property prediction has both advantages and disadvantages. By increasing the number of attributes, the size and dimension of the data increases which can cause problem of over-determining and curse of dimensionality. This problem can be alleviated using PCA. It also helps exclude outliers in the data from our modeling. PCA determines an orthogonal set of axes that lie in the directions of largest, second largest, and so on, down to the smallest variance in the data
- 5 - volume. PCA is of crucial importance because it allows one to recognize the most valuable part of the data. In other words, it determines the directions in which the data has most variations and replaces the original axes of data with new axes which are in the direction of the maximum variances. The directions in which data has minimum variations can be ignored (see Figure 4). Since attributes have different ranges of variation, in order to avoid the dominance of some attributes over the others, attribute data should be normalized so that its mean equals zero and the root mean square (RMS) of all the data is equal to one (RMS=1). Figure 4: Original axes of the data (Attribute1, Attribute2 and Attribute3) are replaced by the directions of first and second maximum variation in the data (shown by two arrows). Therefore dimension of data decreases from 3 to 2. -Extracting Statistical Relationships between Seismic Attributes and Well Logs: Predicting rock properties from seismic attributes is a multi-dimensional curve fitting problem. Even after PCA which decreases the dimensionality of the data, we still deal with a multi-dimensional problem. In dealing with nonlinear and multidimensional problems, it is convenient to divide it into smaller sub-problems which are easier to solve. Figure (5) shows division of the complex 3D surface into smaller sub-planes. There are two main steps for doing this; first step is to classify (cluster) the input data (seismic attributes) and the second step is to compute weights that relate input to output (log data). There are several techniques for classifying the attributes. In this study we have used fuzzy self organizing maps (Eftekharifar et al., 2009, Taner, 1997, Kohonen, 1989, Gustafson and Kessel, 1979). Figure 5: Solving a multi-dimensional curve fitting problem with local linear modeling: Problem of fitting a 3D complex surface is divided into three sub-planes (simpler problems). Nelles, 2001.
- 6 - After clustering the attribute data, the cluster centers are used in Local Linear Modeling algorithm (Eftekharifar, 2009, Nelles, 2001) which determines the weights relating attributes to log data. Results Local Linear Modeling finds the weights relating seismic attributes to well log data. Using blind well test (cross-validation), an iterative way is used for finding the optimum weights that best map log information from seismic data. Once the best weights are found, 3D volumes rock properties can be calculated from 3D seismic data. Figure (6) shows the results of blind well test for modeling a- density (with correlation coefficient of 85%), b- AI (with correlation coefficient of 87%), c- Sonic (with correlation coefficient of 84%) and d- GR log (with correlation coefficient of 73%). The modeled logs are shown in blue and the actual measured values are shown in red. Horizontal axes show the log values and vertical axes show sample number (depth). After finding the optimal model parameters, they can be used to map rock property volumes from 3D seismic data. Some modeling results are shown in Figure (7); a) shows seismic section which was selected for modeling. It shows a section of about 250 ms length, b) shows the result of density modeling where density has been modeled for the entire seismic section and actual log measurements (at location of 50 th trace where a black arrow points to the location of well) are superimposed on the modeled background. As shown in section b, our predictions in background are in an excellent agreement with the actual superimposed log measurements and high/low density layers are clearly distinguishable. Color bar shows the values of density in gr/cc. Section c) shows AI section and the measured AI is superimposed. Color bar represents AI values in units of kg/sm2. Section d) shows the modeled Gamma Ray (shaliness) values for the entire 2D section, again an excellent agreement and consistency is seen between modeled background GR values and superimposed measured GR values at well location (~ 50 th trace). Color bar shows GR values in units of API.
- 7 - a Figure 6: Blind well test (cross-validation) results for modeling a) density, b) AI, c) Sonic and d) GR log, with correlation coefficients of 85%, 87%, 84% and 73% respectively. Horizontal axes show the log values and vertical axes show sample number (depth). b c d Figure 7: a) seismic section used for modeling log properties, b) modeled density log values in gr/cc units, actual measured well log values are superimposed at around 50 th trace. c) modeled AI values in units of kg/sm2 with measured values superimposed at trace 50. d) Modeled GR log section in units of API with measured values superimposed at 50 th trace. More explanations in the text.
Interpretation and Uncertainty Analysis: 3D petrophysical modeling - 8 - In geophysics, every modeling result needs to be interpreted and calibrated based on a priori knowledge which is available through lab or well log analysis. Furthermore, one must determine the uncertainty in any model prediction. At well locations, the model performance can be checked by blind well test as shown in figure (6) but for locations with no well control, we propose an approach based on the rock physics and petrophysics constraints. It is assumed that every rock type in a particular region (local scale) has petrophysical properties in ranges which are defined using available measured data. Therefore our predictions will be valid only if they fall in the pre-defined ranges (templates) and uncertainty in predictions will increase as the predicted data points become more scattered. Uncertainty of predictions can be determined in 2 ways: 1-Uncertainty of prediction itself (regardless of what rock type it is) is determined by the degree of randomness of predictions. That means the more concentrated (less distributed) predictions are more reliable and vice versa. Figure (8) shows an example of these situations where purple predictions which are more concentrated are more reliable than yellow and green predictions. Yellow prediction in turn is more reliable than the green one because it is compact in two directions and only more distributed in one direction (sonic log) whereas the green prediction is widely distributed in all three directions. Randomness is defined by equation (1) which is the sum of Euclidian distances of predicted data points from the center of cluster that they make. where: (1) R: randomness of predictions i=1,2,3,, N : number of log types that are modeled j=1,2,3, M : number of predicted data points Log: Predicted data points Log: Center of cluster of predicted data points 2-Uncertainty of having a desired rock type for a set of predictions is defined by deviations of the predictions from templates. That is, the Euclidian distance between the center of cluster of predictions from the center of template for that desired rock type (obtained from measured data). Figure (9) shows the distance between the center of rock physics template for the desired rock type (for example sand which is shown in blue) and center of predictions (green) Values for sand template are not real. This distance can be quantified using equation (2): where: (2) d: measure of likelihood of having desired rock type i=1,2,3,, N : number of log types that are modeled j=1,2,3, M : number of predicted data points Log: Predicted data points L: Center of cluster for desired rock type template
- 9 - Figure (10) shows a real example of interpretation/ uncertainty analysis where predictions at two different locations of seismic data (first data set) are plotted in 3D (sonic, density and gamma ray log). As seen, the predictions clearly are separated into two main clusters (sand and shale). Figure 8: Distributions of predictions indicate their validity. More distributed predictions are less reliable and vice versa. Figure 9: Uncertainty of having a desired rock type (sand for instance) is defined by the distance of our predictions (green) from the predefined RPT for sand (blue). The values for sand template are not real. Figure 10: Real example from first data set showing the cross plot of predictions. Two lithologies are clearly distinguishable. Conclusions and future work A statistical method has been developed for estimation of rock properties from seismic data and tested successfully on real data sets. Unlike conventional methods that need many wells for reservoir modeling, the proposed method works with limited well log information (one well).
- 10 - Rock Physics Templates are useful tool for interpretation of rock physics predictions. Especially in our case those templates that take into account clay content, porosity and compaction are very effective for distinguishing different lithologies. Joint uncertainty analysis (considering multiple predictions altogether) is a more reliable way to quantify uncertainty at places with no well control. Due to lack of core data in the first data set which was used for inversion of petrophysical parameters from seismic, a more complete data set (Teapot dome data which contains core data), is considered for future research. Also different RPTs will be tested to investigate their applicability for more accurate interpretation and uncertainty analysis. Acknowledgements Many thanks to sponsors of Fluids and DHI consortium for their continuous support and DOE and RMOTC for providing the data. Also we thank Dr. John P. Castagna and our friends in RPL for their help and comments. References Andersen, C. F. and A. V. Wijngaarden, 2007, Interpretation of 4D AVO Inversion results using Rock Physics Templates and Virtual Reality visualization, North Sea examples: 77th Annual International Meeting, SEG, Expanded Abstracts, 77, 2934-2938. Avseth, P., A. V. Wijngaarden, G. Mavko, and T. A. Johansen, 2006, Combined porosity, saturation and net-to-gross estimation from rock physics templates: 76 th Annual International Meeting, SEG, Expanded Abstracts, 76, 1856-1860. Draper, N. R., and Smith, H., 1966, Applied regression analysis: John Wiley & Sons, Inc. Eftekhari Far, M., Extrapolation of log properties by Integrating F-SOM and Local Linear Modeling. Annual SEG annual meeting Exp. Abs., Houston, 2009. Eftekharifar, M, M. A. Riahi, R. Kharat, 2009, Integration of Gustafson- Kessel algorithm and Kohonen s Self Organizing Maps for Unsupervised Clustering of Seismic Attributes, Journal of Seismic Exploration, Volume 18, Number 4, October. Gustafson, D. E. and W. C., Kessel, 1979, Fuzzy clustering with a fuzzy covariance matrix. In: Proc IEEE CDC, San Diego, CA, USA, pp. 761-766. Hampson, D. P., J. S. Schuelke, and J. A. Quirein, Use of multi-attribute transforms to predict log properties from seismic data, Geophysics, 66, 220-236, 2001. Hashin, Z. and S. Shtrikman, 1963, A variational approach to the elastic behavior of multiphase material: J. Mech. Phy. Solids., 11, 127-140. Kohonen, T., 1989, Self-organization and associative memory, 3rd ed.: Springer-Verlag.
- 11 - Marion, D. P., 1990, Acoustic, mechanical, and transport properties of sediments and granular materials: Ph.D. dissertation, Stanford University. Mindlin, R.D., 1949, Compliance of elastic bodies in contact: J. Appl. Mech., 16, 259-268. Nelles, O., 2001, Nonlinear system identification: Springer-Verlag. Ǿdegaard, E. and P. Avseth, 2004, Well log and seismic data analysis using rock physics templates: First Break, 23, 37-43. Schultz, P. S., Ronen, S., Hattori, M., and Corbett, C., 1994, Seismic guided estimation of log properties, parts 1, 2, and 3: The Leading Edge, 13, 305 310, 674 678 and 770 776. Taner, M.T. 1997, Kohonen s Self Organizing Network with Conscience, Rock Solid Images Technical Report. Taner, M.T., Koehler, F., and Sheriff, R.E., 1979, Complex seismic trace analysis: Geophysics, 44, 1041-1063. Taner, M. T., Schuelke, J. S., O Doherty, R., and Baysal, E., 1994, Seismic attributes revisited: 64th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 1104 1106. Table 1: Production details of Tensleep reservoir