Topological Data Analysis of Marcellus Play Lithofacies

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1 URTeC: Topological Data Analysis of Marcellus Play Lithofacies Andrea Cortis, AYASDI, Menlo Park, CA, USA Copyright 2015, Unconventional Resources Technology Conference (URTeC) DOI /urtec This paper was prepared for presentation at the Unconventional Resources Technology Conference held in San Antonio, Texas, USA, July The URTeC Technical Program Committee accepted this presentation on the basis of information contained in an abstract submitted by the author(s). The contents of this paper have not been reviewed by URTeC and URTeC does not warrant the accuracy, reliability, or timeliness of any information herein. All information is the responsibility of, and, is subject to corrections by the author(s). Any person or entity that relies on any information obtained from this paper does so at their own risk. The information herein does not necessarily reflect any position of URTeC. Any reproduction, distribution, or storage of any part of this paper without the written consent of URTeC is prohibited. Summary We present a study on the lithofacies characterization of the Marcellus shale gas formation. The data set consists of nine vertical wells, each with petrophysical logs of composition (quartz, calcite, clay, and total organic carbon) and elastic parameters (density, and compressional and shear velocity) along four geological sections, i.e., the Mahatango Formation, and the Upper Marcellus, the Cherry Valley carbonate formation, and the Lower Marcellus members of the Marcellus Formation. We successfully used a new mathematical technique known as Topological Data Analysis (TDA) to identify lithofacies groups in the vertical profiles that possess well-defined marginal distributions in the velocity-density plane. Motivation The definition of lithofacies in a vertical log profile is a complex multidisciplinary task that often rests on the professional experience of groups of petrophysicists, geophysicists, geologists, and sequence stratigraphers. Statistical and machine learning data-mining methods such as Principal Component Analysis, k-means and hierarchical clustering, Gaussian Mixture models, and Neural Network Classifiers are often used to make sense of vertical log profiles data. These methods, however, often produce less than satisfactory predictions and we find ourselves in want of more efficient and rigorous mathematical methods. In this work, we consider nine vertical wells taken from the Marcellus gas shale play along four geological sections, i.e., the Mahatango gray shale, the Upper Marcellus, the Cherry Valley carbonate formation, and the Lower Marcellus. The petrophysical logs for these wells include composition (quartz, calcite, clay, and total organic carbon content) and elastic parameters (density, and compressional and shear velocity). Figure 1 shows the total organic carbon (TOC) content vertical profiles for these nine wells. Note that the absolute scale for the TOC content and the absolute top and thickness for each of the wells has been removed to ensure data anonymity. From the TOC vertical profiles, however, it is straightforward to identify the location of each of the producing layers. The total number of variables is seven (four for composition and three for elastic properties) and the total number of measurements for all the wells is 18,650.

2 Figure 1: Total organic carbon (TOC) profiles for the nine wells subject of this investigation. The actual depth and thickness of the four layers for each of the nine wells has been obfuscated. It is often believed that this type of data should cluster in the acoustic impedance (AI) vs poisson ratio (PR) plane, having well-defined marginal distributions. In Figure 2, we show how that the data we are considering in this contribution plot in such a plane. Following Wang and Carr (2014), three geologically based groups have been identified, based on their clay and quartz-to-calcite-ratio (QCR) values: GSS: a clay-poor (<0.4) QCR-rich (>3) gray siliceous shale (GSS), GMS: a clay-poor (<0.4) QCR-intermediate (>1/3, <3) gray mixed shale (GMS), GMD: a clay-rich (>0.4) gray mudstone (GMD). Figure 2: (Bottom Left) Crossplot of acoustic impedance (AI) vs poisson ratio (PR) for three distinct lithological groups, and associated marginal distributions. (Top Left) Marginal distribution of the three groups from the bottom left cross plot, they do not exhibit significant differences and are therefore indistinguishable for practical purposes. (Top Right) Associated dendrogram representation of the three groups. (Bottom Right) Marginal distribution of the three groups from the top right dendrogram do not exhibit significant differences and are therefore indistinguishable for practical purposes.

3 As it can be readily observed from Figure 2, the groups proposed by Wang and Carr (2012) do not separate out in the AI-PR plane. Also, the dendrogram representation of the data (Figure 2 - top right) does not provide any further insight as to which is the best group definition. An objection towards this type of classical petrophysical analysis is such that the boundaries of the groups need to be specified a priori, based on extensive knowledge of the data that sometimes cannot be transferred from one play to another. Once the first set of hypothesis has failed, one has to look at another set of hypothesis, for both the group boundaries and the chosen representation, making the task tedious and with no guarantees of convergence towards one good solution. Thus, it would be desirable if the data itself could suggest the correct groups and the best representations. This is, in fact, the research program of the mathematical technique known as Topological Data Analysis (TDA). Instead of defining ad-hoc hypotheses and iteratively code and test models, TDA provide a means to group data together in a logical way, almost agnostic to the subject matter. We refer the mathematically inclined reader to Carlsson (2014) for a recent overview of the subject. A growing number of large companies and institutional organizations are successfully using TDA on a wide variety of data-driven problems as they arise in genetics, pharmaceutics, healthcare, finance, energy, security, and defense. To our knowledge, this is the first time that TDA has been applied in the literature to the solution of a geophysical problem. In the following, we will discuss how and why TDA improves on existing data discovery techniques, and show an application to a set of nine petrophysical well logs typical of the Marcellus shale gas play. Topological Data Analysis: a short introduction TDA uses topology, the mathematical study of geometrical shape, to understand complex datasets. Linear regression, the fitting of a straight line to a cloud of points on a plane, is perhaps the most rudimentary type of topological data analysis all scientists and engineers are familiar with. A cloud of points on a plane roughly distributed along a circle is another familiar shape that is associated with periodicity. Groups of points on a plane also have a topological interpretation in terms of disconnected, i.e., independent logical units. Another familiar shape is the Y junction that is associated with bifurcation phenomena. When the number of dimensions of the data increases, however, it becomes increasingly difficult to discern geometrical features such as these and it becomes all the more difficult to derive locally linear models. TDA builds upon and generalizes these geometric concepts to real world datasets that can span millions of explanatory variables and millions of measuring points. The idea behind the application of TDA is to represent data via topological networks, i.e., data is represented by grouping similar data points into nodes, and connecting the nodes by an edge if the corresponding nodes have at least a data point in common. Because each node does represent multiple data points, the network gives a compressed, low dimensional version of extremely high dimensional data. When used as a framework in conjunction with machine learning, TDA enables the understanding of the shape of complex data sets, highlighting previously hidden groups of data, and revealing the relevant explanatory variables. TDA starts by projecting the data onto a low dimensional space via one or more lens functions. Familiar examples of such lens functions are, for instance, the main components of Principal Component Analysis, a statistical measure along the various variables, or even the value of a data variable itself. Projections onto the outcomes of numerical models are also possible. The projected points are then covered by overlapping sets in this low dimensional space and the amount of overlap contributed defines the shape of the network. No overlap will result in a completely disconnected set of points, while complete overlap will result in a single group of data, none of which are desirable outcomes for a data analysis. The points in each of the overlapping sets are then clustered in the original data space to create nodes, i.e., agglomerates of similar points. Similarity between the points is defined via the choice of a metric function and the granularity of the nodes is regulated by a resolution parameter.

4 The absolute position of the nodes in the graphical representation of the network is irrelevant and the network can be squeezed and stretched (but not cut) without altering its fundamental properties. Only the relative connection between pairs of nodes, such as the presence or absence of an edge between any two given nodes. This point can be readily understood by reflecting upon the hand example shown in Figure 3. Different choices of the similarity measure among points, the associated resolution parameter, the projecting lens and associated gain overlap parameter all lead to the definition of different networks. Interesting networks are characterized by the separation of node groups, by presence of holes and also flares, i.e., elongated sets of nodes that protrude from otherwise bulky node groups. Some points will not be close to any other points in the dataset and are thus referred to as the outliers. Once a set of lens and metric functions has been defined, we check that the network is stable in the presence of small perturbations of the resolution and gain parameters. Once a suitable network has been defined, we can also define groups of nodes in the network that share similar characteristics. We will refer to these simply as groups. Each group possesses its own distinctive characteristics, which can be determined either by some topological feature, i.e., a flare, an isolated set of nodes, a set of nodes with one or multiple holes in it, or by the statistical characteristics of the variables for all the points that constitute each node. For instance, a group labeled G1 may have a different statistical character in one or more variables with respect to a group labeled G2. Topological Data Analysis: application to petrophysical data We can now apply TDA to the petrophysical data described in the Motivation section, beginning with an 18,650x7 matrix of input data. After experimentation with various choices of metrics and lenses (not reported here), we have found an interesting network by using the first two PCA components with a gain equal to 1.8, and a variance normalized Euclidean metric with resolution equal to 45. The network is displayed in Figure 4 and the nodes have been colored by TOC content. It can be immediately surmised from the network that relatively high TOC measurements cluster together in four distinct groups, labeled A, B, C, and D. We can easily check which variables are responsible by taking the cumulative density distribution of each of the variables for the two groups and calculating the Kolmogorov-Smirnov (KS) statistics. If the value of the KS statistics is sufficiently large for a given variable (>0.65), then this suggests that the variable explains the difference between those two groups. This statistical analysis shows that groups A and B are considered different because B has higher calcite content, 0.38, than A, 0.11, and because B has a lower quartz content, 0.21, than A, Groups B and D show differences in the mean TOC content (0.054 and 0.029, respectively), as it would be the case for a multimodal distribution. The differences in the calcite content and elastic parameters suggest that group D is a more brittle rock than group B. The analysis also suggests that Group C is dominated by brittle rocks, which differs from group D being poorer in calcite and richer in quartz. Lastly, we observe Figure 3: (A) Illustrative representation of the TDA methodology on a cloud of data points representing a hand in three dimensions. (B) Represents the data points by projecting them on a two dimensional plane and coloring by the distance to the wrist. (C) illustrates the overlapping sets binning step, and (D) illustrates the connectivity of the binned points, where edges join overlapping sets. The overall impression of pane D is of the hand s skeleton, which further illustrates why only the connections are important and not the absolute position of the nodes.

5 that the network in Figure 4 also displays a group of isolated points that are outlier measurements. Figure 4: TDA network of the Marcellus nine wells petrophysical logs dataset subject of this paper. The network shows a number of interesting features, such as the flares in the main network body, a number of isolated groups on the top, and a number of outlier measurements. The network is colored by TOC content, and four regions of relatively large TOC content, labeled A-D, can be readily identified by simple visual inspection. Figure 5 displays the same network as in Figure 4, but colored by compressional velocity, v p. We observe two regions, labeled G1 and G2 that do not have any distinctive features when colored by compressional velocity, and yet are separated out by the topology of the network. This is an indication that other variables are responsible for this Figure 5: Same TDA network as in Figure 4, but colored by compressional velocity, v p. Compressional velocity is high in the two regions labeld G1 and G2, even though the two groups of nodes are topologically separated. For an explanation see text and captions for Figures 6 and 7.

6 topological separation. The univariate KS statistical analysis of all variables suggests that calcite and quartz content are responsible for the difference in the two groups, which can verified when the network is colored by calcite (Figure 6) and quartz content (Figure 7). Figure 6: Same TDA network as in Figure 4, but colored by calcite content. Group G1 is rich in calcite, while group G2 is poor in calcite. Figure 7: Same TDA network as in Figure 4, but colored by quartz content. Group G1 is poor in quartz, while group G2 is rich in quartz. In Figure 8, we have uniquely colored each group of nodes in the network, such that the same color is applied to the nodes and points within their respective group. The only points that may have a color ambiguity are those at the boundary between groups. Because of the aforementioned topological construction, the nodes at the boundary between two groups do contain overlapping points if the two groups are joined by one or more edges. In this study

7 we choose to color the boundary points with the group color of the point immediately above, in the vertical section. This is a minor technicality that will not affect the quality of the analysis representation, due to the fact that the point count of the groups interiors greatly outnumbers the amount of boundary points. Figure 8: Same TDA network as in Figure 4, where each node has been colored by groups It is important to note that we can map the group color representation on the TOC vertical profiles (or of any other variable), and see how they represent the lithological stratification. We find that the high TOC groups of Figure 6 (CG52, CG78 and CG109, pink/rose in color), when plotted in the vertical profiles of Figure 9, correspond to the general understanding of the Upper and Lower Marcellus producing layers. The green profile of the nine wells in Figure 9 is characterized by high values of calcite and correlates with the Cherry Valley carbonate layer between the Upper and Lower Marcellus. The green groups/profiles are also observed between the large yellow zones, which characterize the Mahatango gray shale, indicating fine carbonate layers. Figure 9: Same profiles as for Figure 1, but colored according to the groups in Figure 8

8 Figure 10 is a scatter plot of the compressional velocity, vp, and the density, ρ, colored by TOC content. The underlying transparent layer for the panes in Figure 10 shows hexagonal binning of all the data, from which no striking patterns are discerned. The 12 panes also overlay an opaque scatterplot that depicts the TOC content of each group. The panes associated to the groups CG52, CG78, and CG109 correlate to relatively low values of density and compressional velocity, as shown by the overlaid opaque scatterplot points in the lower left part of the panes, which is a distribution expected for soft rocks. Moreover, it can be observed that all the groups plot in different regions of the panes and have very distinguishable marginal distributions. Figure 10: Scatterplots of compressional velocity vs density colored by TOC content. Transparent background layer is common to all panes and represents the hexagonal binning of TOC for all data points. The opaque foreground of each pane shows a TOC scatterplot for the points corresponding to the groups defined in Figure 8. Conclusions While a proven method in the statistics and mathematics communities, TDA now provides a new and improved methodology of lithofacies identification for geoscientists. The application of TDA to well logs in the Marcellus Formation provided clear identification of our four geologic layers of focus for our study. Four groups of relatively high TOC content have been identified by the TDA analysis, and the variables that elicit their differences have been analyzed. All the groups identified by TDA have very clear representations in the compressional velocity - density representation, and include meaningful marginal distributions. Within the limits of the Mahatango gray shale, we were able to identify fine-layered carbonate structures that could not be identified by classical classification

9 methods. Finally, we note that TDA lithofacies classification is not just relevant for shale-gas unconventional exploration, but provides a new means to look at all types of geological structures. This study does not aim to be a complete and universal characterization of the Marcellus lithofacies, but rather it is meant as an explanatory illustration of how the TDA methodology can be critical in identifying and properly characterizing interesting lithofacies. More careful investigations would naturally include a larger number of wells, and more detailed characterization of all the network groups, all of which is clearly beyond the scope of the present contribution. In conclusion, TDA is a very general and powerful tool, which can be applied to the analysis of any complex data sets in the oil and gas industry. It proves to be especially useful when the identification of the main variables is too complicated to achieve by traditional first-principle methods. References Carlsson, G. "Topological pattern recognition for point cloud data." Acta Numerica 23 (2014): Wang, G., and Carr, T.R. "Methodology of organic-rich shale lithofacies identification and prediction: A case study from Marcellus Shale in the Appalachian basin." Computers & Geosciences 49 (2012):