A methodology for the characterization of flow conductivity through the identification of communities in samples of fractured rocks
Introduction
Many real world problems such as biological, social, metabolic, food, neural networks and pathological networks among others can be modeled and studied as complex networks (Kolaczyk, 2009, Cohen and Havlin, 2010, Estada, 2011). They are mathematically represented and topologically studied to uncover some structural properties. In the petroleum industry one issue of importance is the study and analysis of fluid flow in fractured rocks. In this paper we present a methodology for the characterization, through the topology of a network, of the capability of flow conductivity in fractures associated to a reservoir under study. Our methodology extracts from a fracture image a graph focusing on two key aspects. The first is to identify regions of fractures that are more conductive, and the second one is to find nodes that belong to the largest paths that have more possibility of serving as flow channels. This paper deals with real fracture networks derived from original hand-sample images. These images of rocks correspond to a Gulf of Mexico oil reservoir, and are used as test examples for identifying properties related to the fluid flow from a topological perspective. This methodology assumes that the fractures in the image have all being identified as conductive. Then it determines qualitatively different conductive regions in the fracture network through the analysis of the cross points of the fractures, and quantifies the connectivity among these cross points and their topological function within the network. This methodology consists of: (i) the application of centrality measures that involves the estimation of shortest paths, and (ii) the identification of node sets by means of community detection. The communities are subunits associated with the more highly interconnected parts used for determining the global organization in the network (Lancichinetti, Kivelä, Saramäki, Fortunato, 2010). Many methods have been developed for the identification of communities (Clauset et al., 2004, Girvan and Newman, 2002, Newman, 2004, Porter et al., 2009, Radicchi et al., 2004). We apply an efficient method reported in the literature (Condon and Karp, 2001, Lancichinetti and Fortunato, 2009) for grouping sets of nodes based on a modularity function. In addition, for the construction of these communities a formulation for computing the weights among cross points is proposed. This approach will help in analyzing different study regions and to characterize the fracture networks by means of the topological properties obtained, and hence it can identify conductive regions. Also these results can be used in combination with other geophysical or petrophysical properties from the fracture network.
The paper is organized as follows. In Section 2, previous work and basic concepts are described; in particular the centrality measures and a method for determining communities or regions are discussed. In Section 3, our general scheme is explained. In this part the association between the centrality measures and the identification of communities are described. In Section 4, we show our results, applying the methodology to fracture hand-sample images. Finally, in Section 5, we give our conclusions.
Section snippets
Previous works and theoretical framework
In the characterization of naturally fractured reservoirs (NFR) one of the main challenges in the hydrocarbon industry is the generation of a representative model for it (Aguilera, 1995, Baker, 2000, Narr et al., 2006, Nelson, 2001, Nikravesh, 2004). This characterization requires putting together different data sources about the whole reservoir (Bogatkov and Babadagli, 2007, Gauthier et al., 2002, Guerreiro et al., 2000). One of the important problems is the determination of the nature, and
Identification of conductive regions
The process of characterizing regions leads to analyze the capability to conduct fluid in the fracture network; here it is described in three steps represented in the second dashed box in Fig. 1. The first box includes the image processing of fractures, whose detailed description is presented in Santiago et al. (2012). In this paper, we describe the details of the second box.
Results
Other fracture network belonging to the same Cantarell reservoir is shown in Fig. 8. Note here the presence of many small components, even isolated fracture segments. The so-called component determination, gives us an indicator of the disconnectedness in the fracture network. The application of centrality measures helps to identify the location of the largest component. This fracture image has 234 cross points, 17 components, and 35 communities (weighting by distances). Results of the
Conclusions
In this work, a general methodology for the analysis of conductivity of fluid flow in fracture networks is presented using a complex network approach. The input data are networks generated from real images of fractured rocks where we assume that all fractures are conductive. More detailed geological analysis is necessary to determine the true nature of the fractures (either conductive or not). Nevertheless our methodology does not depend on this detail. Provided we have a fracture network the
Acknowledgements
This work was supported by SENER-CONACyT grant 143935 under project Y.00114 of the Mexican Petroleum Institute (IMP). We thank the fruitful discussions and insights of Dr. Luis G. Velasquillo, Dr. Ildar Batyrshin, Dr. Diego Del Castillo and Dr. Ernesto Estrada.
References (44)
Naturally fractured reservoirs
(1995)The rush in a graph
(1971)- Baker, R. O., & Kuppe, F. (2000). Reservoir characterization for naturally fractured reservoirs. In Paper SPE 63286,...
- Bastian, M., Heymann, S., & Jacomy, M. (2009). Gephi: An Open Source Software for exploring and manipulating networks....
A mathematical model for group structure
Applied Anthropology
(1948)Characterizing flow and transport in fractured geological media: A review
Advances in Water Resources
(2002)- et al.
Fast unfolding of communities in large networks
(2008) - Bogatkov, D., & Babadagli, T. (2007). Characterization of fracture network system of the midale field. In Paper...
A faster algorithm for betweenness centrality
Journal of Mathematical Sociology
(2001)- et al.
Modeling fracture flow with a stochastic discrete fracture network: Calibration and validation
Water Resources Research
(1990)
Finding community structure in very large networks
Physical Review E
Complex networks: Structures, robustness and function
Algorithms for graph partitioning on the planted partition model
Random Structures & Algorithms
Improved spectral algorithm for the detection of network communities
Semantic image interpretation of gamma ray profiles in petroleum exploration
Expert Systems with Applications
Community detection in graphs
Physics Reports
A set of measures of centrality based on betweenness
Sociometry
Integrated fractured reservoir characterization: A case study in a North Africa field
SPEREE
Fluid flow complexity in fracture network: Analysis with graph theory and LBM
Community structure in social and biological networks
Proceedings of the National Academy of Sciences of the United States of America
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