A methodology for the characterization of flow conductivity through the identification of communities in samples of fractured rocks Elizabeth Santiago ⇑ , Jorge X. Velasco-Hernández, Manuel Romero-Salcedo Programa de Matemáticas Aplicadas y Computación, Instituto Mexicano del Petróleo, Av. Eje Central Lázaro Cárdenas Norte, 152. Col. San Bartolo Atepehuacan, Mexico-city CP 07730, Mexico article info Keywords: Complex networks Centrality measures Network communicability Network topology Naturally fractured reservoir abstract We present a methodology that characterizes through the topology of a network the capability of flow conductivity in fractures associated to a reservoir under study. This strategy considers the fracture image as a graph, and is focused on two key aspects. The first is to identify communities or sets of nodes that are more conductive, and the second one is to find nodes that form the largest paths and have therefore more possibility of serving as flow channels. The methodology is divided into two stages, the first stage obtains the cross points from fracture networks. The second stage deepens on the community identification. This second stage carries out the process of identifying conductive nodes by using centrality measures (betweenness, eccentricity and closeness) for evaluating each node in the network. Then an optimization modularity method is applied in order to form communities using two different types of weights between cross points or nodes. Finally, each community is associated with the average value of each measure. In this way the maximum values in betweenness and eccentricity are selected for identifying communities with the most important nodes in the network. The results obtained allow us to show regions in the frac- ture network that are more conductive according to the topology. In addition, this general methodology can be applied to other fracture characteristics. Ó 2013 Elsevier Ltd. All rights reserved. 1. 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, & Havlin, 2010; Estrada, 2011). They are mathemat- ically 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 character- ization, 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 conduc- tive 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, Newman, & Moore, 2004; Girvan & Newman, 2002; Newman, 2004; Porter, Onnela, & Mucha, 2009; Radicchi, Castellano, Cecconi, Loreto, & Parisi, 2004). We apply an efficient method reported in the literature (Condon & Karp, 2001; Lancichinetti & Fortunato, 2009) for grouping sets of nodes based on a modularity function. In addi- tion, 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 geo- physical or petrophysical properties from the fracture network. 0957-4174/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.eswa.2013.08.011 ⇑ Corresponding author. E-mail addresses: esangel@imp.mx (E. Santiago), velascoj@imp.mx (J.X. Velasco- Hernández), mromeros@imp.mx (M. Romero-Salcedo). Expert Systems with Applications 41 (2014) 811–820 Contents lists available at ScienceDirect Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa