An Integrated Strategy for Analyzing Flow Conductivity

of Fractures in a Naturally Fractured Reservoir using a

Complex Network Metric

Elizabeth Santiago, Manuel Romero-Salcedo, Jorge X. Velasco-Hernández, Luis G.

Velasquillo, J. Alejandro Hernández

Instituto Mexicano del Petróleo, Av. Eje Central Lázaro Cárdenas Norte, 152

Col. San Bartolo Atepehuacan, Mexico-city, Mexico CP 07730

{esangel,mromeros,velascoj,lgvelas}@imp.mx

jahernandezv87@gmail.com

Abstract. In this paper a new strategy for analyzing the capability of flow

conductivity of hydrocarbon in fractures associated to a reservoir under study is

presented. This strategy is described as an integrated methodology which

involves as input data the intersection points of fractures that are extracted from

hand-sample fracture images obtained from cores in a Naturally Fractured

Reservoir. This methodology consists of two main stages. The first stage carries

out the analysis and image processing, whose goal is the extraction of the

topological structure from the system. The second stage is focused on finding

the node or vertex, which represents the most important node of the graph

applying an improved betweenness centrality measure. Once the representative

node is obtained, the intensity of intersection points of the fractures is

quantified. In this stage a sand box technique based on different radius for

obtaining an intensity pattern in the reservoir is used. The results obtained from

the integrated strategy allow us to deduce in the characterization of reservoir,

by knowing the possible flow conductivity in the topology of the fractures

viewed as complex network. Moreover our results may be also of interest in the

formulation of models in the whole characterization of the reservoir.

Keywords: flow conductivity of fractures, complex network metric, naturally

fractured reservoirs.

1 Introduction

One of the principal challenges in characterization of naturally fractured reservoirs

(NFR) in the hydrocarbon industry is the generation of a representative model of the

reservoir [1, 2, 3, 4]. This characterization requires putting together different data

sources about the whole reservoir [5, 6, 7]. One of the most important problems is the

determination of the nature, and disposition of heterogeneities that inevitably occurs

in petroliferous formations in order to determine the capability for fluid transport.

Different strategies have been developed to tackle this problem. Some authors have

2

focused on the analysis of the properties of the fluid flow [8, 9], others in the

modeling and simulation of fracture networks [10], and some others in the analysis of

topological properties by applying statistical techniques to the structure where the

fluid transport may occur [11]. In this last approach most authors use synthetically

generated fracture networks [26]. This paper deals with this last approach, but through

the extraction of parameters from original hand-sample images supplied by

geologists. These images correspond to a Gulf of Mexico oil reservoir, and will be

used as test examples for the determination of network topologies [12, 13]. In

particular, we present and discuss the application of a network metric, which

computes the number of the shortest paths that pass through a certain node [13]. In

this work, a new strategy for the identification of patterns on a set of fracture images

is presented. The processing and analysis of fracture hand-sample images uses the

methodology KDD (knowledge discovery in database) [14, 27] in order to find

patterns embedded in their topologies. We are particularly interested in the

importance of a node as related to its topological function within the network. The

methodology includes a metric designed to characterize and identify “an important

node”, and with it a new strategy is implemented to qualitatively asses flow capability

from the fracture images. The metric used here applies the notion of the vertex’s

importance in a graph that depends on many factors to be described later in this work.

Finally, a step of evaluation of this node is carried out to estimate the intensity of

intersection points of fractures.

The paper is organized as follows. In section 2, the general scheme of the proposed

methodology is described. The image preprocessing is also treated and the applied

methods are explained. In section 3, the analysis of nodes is evaluated and the

description of the improved betweenness centrality metric used is presented. In

section 4, we show our results, first applying the methodology to a set of four fracture

hand-sample images, and then applying it to a set of 100 fracture hand-sample

images. Finally, in section 5, the conclusions and future work are presented.

2 New Methodology for Characterizing the Topology of

Fractures in NFR

In the next paragraphs, a novel methodology for the analysis of the topological

structure of fracture networks is applied. Our input data are fracture images. The

preprocessing and analysis of fractures is applied to a set of 100 hand-sample fracture

images of cores (samples of rocks recovered from a formation of interest commonly

used for its study and evaluation) of the Gulf of Mexico. These images have a bmp

format. Some examples (labeled as Frac-6 and Frac-93) are shown in Fig. 1. Most of

the software available for analyzing the topology of complex networks (such as

Gephy and Cytoscape) require as input data files that store the adjacency matrix of the

graph. Our methodology generate the adjacency matrix through several steps to

recover the fracture structure from the original fracture images, and then the

application of the necessary operations to characterize intensity patterns of non-

terminal nodes (or intersection points) in fractures. We define an intensity pattern as a

3

tendency in the increase of number of nodes in different sizes of circles. The general

procedure is described below.

a) Frac-93

b) Frac-6

Fig. 1. Hand sample examples of fractures associated to NFR: a) Frac-93, and b) Fract-6 from

Jujo-Tecominoacán reservoir.

Considering the aforementioned observations, our strategy for identifying patterns

in the distribution of the fracture intersections is presented in two stages: 1) creation

of a graph from the processed image, and 2) analysis of the intensity of nodes in the

fractures. Both are outlined in Fig. 2. The description of the first stage is explained

below, and the second stage is detailed in section 3.

Fig. 2. General schema of the proposed methodology

4

The goal of the first stage is to obtain the adjacency matrix representative of the

topological structure of the fracture network. We do it in three steps. In the first step

the image is processed, applying a skeletonization process for recovering the structure

of the fracture in the whole hand-sample image. The second step identifies and

obtains the intersections and endpoints of each fracture image, what are known in

graph theory as non-terminal and terminal (leaves) nodes, respectively. The third step

generates a file that contains the adjacency matrix, where the recovery of all the nodes

and edges is done. It requires touring all the paths in the skelotonized fracture in order

to find the neighborhood nodes, i. e., to identify which nodes are linked to each node

for establishing the connection. Then, this matrix is used for building the graph. The

second stage indentifies patterns in the distribution of intersection points in the

fractures by means of the application of a centrality metric widely used in complex

networks. In the next section, the main idea and the definition of this metric are

explained.

3 Fracture Intensity Analysis

The second stage is shown in the dotted lower rectangle in Fig. 2. This stage is

subdivided into three steps. An important property of the fracture network is the

capability of transport fluids. In this case, the non-terminal nodes or intersections of

traces are considered. Thus, the strategy is to analyze different sizes of circular

regions from a particular node for evaluating precisely the amount of these nodes in

each region. It will allow us to identify any pattern in the increment of different sizes

of radius considering the number of intersection points computed. The first step of the

second stage is to select the most important node from the set of nodes stored in the

adjacency matrix. The earliest intuitive conception of point centrality was based upon

the structural properties of centrality. This idea was introduced by Bavelas [15], and

an essential tool for the analysis of networks is the centrality index which is based on

counting paths going through a node [16]. These measures define centrality in terms

of the degree to which a point falls on the shortest path between others, and therefore

it has a potential for controlling the communication of the network. One of these

metrics is the betweenness centrality measure which was proposed by Freeman [17],

and Antonisse [18]. In this work, an improved version [19] of this measure is applied,

which includes a more efficient and faster algorithm for large and very sparse

networks. This algorithm is based on an accumulation technique that solves the

single-source shortest-path problem, and thus exploits efficiently the sparsity of the

network incidence matrix. The traditional formulation for computing the betweenness

centrality index is accomplished in two steps: i) compute the length and the number of

shortest paths between all pairs, and ii) sum over all the pair-dependencies. For each

node i, in the network, the number of “routing” paths to all other nodes (i.e., paths

through which exist connectivity) going through i is counted, and this number

determines the centrality i. The most common index is obtained by taking only the

shortest paths as the routing paths. Formally the definition of betweenness centrality

of a node i, is given by (1):

5

where stands for summing each pair once, ignored the order, and equals

1 if the shortest path between nodes j and k passes through node i, and 0 otherwise. In

networks with no weight, i.e., where all edges have the same length, there may be

more than one shortest path. In this case, it is common to take , where is the number of shortest paths between j and k, and

is the number of those going through i. An improvement to this equation is

presented in (3), where the numerator of the pair-dependency in (2), , is

obtained by the Bellman criterion, , if the shortest paths between j and

k pass through i [19]. A high centrality score indicates that a vertex can be reached by

others on short paths, or that a vertex connects to others. In the methodology

proposed, this metric is applied for all the non-terminal nodes of each fracture image,

where the maximum value obtained is considered as the main node in the fracture

system.

The second step in stage two consists in the computation of the intensity of non-

terminal nodes that connect fractures. The idea is to carry out a sampling of the total

image area with concentric circles centered at a strategic node, in this case, taking the

most important node chosen by the betweenness centrality measure, and then to count

the number of non-terminal nodes comprised in each circle. This process is described

as follows. In Fig. 3 the intersection points among fractures are computed and

presented, and the metric of centrality is done (see Fig. 4 and Fig. 5). Once the most

important node in a graph is identified, it will be considered as the origin for the

sampling circles. The third step is to draw the tendencies of all non-terminal nodes.

4 Results

In experiment, 100 hand-sample fracture images of the Gulf of Mexico oil Reservoir

are considered. The images used were provided by experts of the geological area.

Applying the first stage of the strategy (see Fig. 2), all the images are processed for

recovering their fracture structure, in most cases required of the skeletonization

process. For convenience only two images are shown in Fig. 1. Fig. 3 presents the

resulting images after the application of the second step concerning to the

identification of the non-terminal nodes or intersection points related to the images in

. (1)

. (2)

if

otherwise (3)

6

Fig. 1, where the number of nodes is listed at the bottom of each image (61and 252

non-terminal nodes which correspond to Frac-6, and Frac-93, respectively). The next

step is the determination of the topological structure of the fractured network

employing the non-terminal nodes and the distance among their neighbors for its

building. For drawing the resultant graph, the Cytoscape tool [20, 21] has been used,

receiving as input data the adjacency matrix generated in the previous steps, this is

carried out for each hand-sample image. The resultant graphs of Frac-6 and Frac-52

(see Fig. 4 and 5, respectively) are represented using a hierarchical structure. The

number of conex components in Fract-6 is 17 and 30 conex components in Frac-93,

which they represent the connectivity of the fractures studied.

In the second stage, for obtaining the main node, first the improved betweenness

centrality measure is applied by using the Gephy tool [22, 23]. Then, the maximum

value obtained using this metric is selected from all nodes in the fracture, and this

maximum value of the corresponding node is considered the main node. To be precise

a main node is defined as a node that has the highest connectivity according to the rest

of the nodes in the fracture system, for this problem, the largest capability of transport

fluids. This procedure was repeated for each fracture image. In Fig. 4 and 5, the main

nodes are highlighted in black color, they correspond to nodes 7 and 25 in Frac-6, and

node 76 in Frac-93; their betweenness centrality values are 195 and 17567.1,

respectively. This operation was applied to the 100 test instances. In Table 1 only 20

fracture images are shown, where the first column presents the label of the image, the

second column indicates the label of the main node obtained by applying the

centrality metric, and the third column has the total number of non-terminal nodes

analyzed. It is important to comment that some fractures obtained two main points,

for example Frac-6 has 2 main nodes (7 and 25) since both have the same

betweenness centrality measure.

Once the main node of each fracture is found, the next step is the determination of

the distribution of the nodes in the whole network. In this experiment, we take the

node or nodes identified as “main nodes” according to the definition given above, and

use them as centers or concentric circles of increasing radius. The radii of the

sampling circles is increased (arbitrarily) in 10 pixels, taking as origin the main node

previously found. For instance, in Fig. 6a and 6b, the scaling of radius is shown,

where the main nodes 25 and 76 were taken from Table 1 (corresponding to Frac-6

and Fract-93, respectively). Considering the node-25 of Frac-6, an example of growth

of the number of non-terminal nodes in circular areas is presented in Table 2. The

initial size of the radius is 10 pixels, and the next circles were generated in augment of

10 pixels, i.e., 20, 30, 40 pixels and so on until covering all the non-terminal nodes in

the fracture image. In this instance, 84 regions were produced for covering the 61

nodes initiating in node-25 of Frac-6. The empirical cumulative distribution of the

nodes of each image is shown in Fig. 7. In this figure each line represents the increase

of the intersection points (non-terminal nodes) of each of the 100 fracture images, the

axis of the coordinate is the distance of the radius in pixels, the axis of the abscise is

the amount of the nodes found in each region, and the circle at the end of the line

indicates the farthest node from the main node. Note in this figure that a general

description from the set of 100 images two groups can be identified: one group where

7

the intersection points of fractures (non-terminal nodes) are more scattered, and the

second group where the intersection points are nearer among them.

a) Frac-6

b) Frac-93

Fig. 3. Identification of intersection points (non-terminal nodes)

Fig. 4. Resultant graph of Frac-6

Frac-93.

Number of Intersection: 252

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8

Fig. 5. Resultant graph of Frac-93

Table 1. Result of the betweenness centrality metric in the test set

Fracture

label

Main node

label

Total of non-

terminal

nodes

Fracture

label

Main node

label

Total of non-

terminal

nodes

Frac_(3) node-15 24 Frac_(80) node-13 107

Frac_(6) node-7 61 Frac_(82) node-28 64

Frac_(6) node-25 61 Frac_(83) node-47 107

Frac_(9) node-34 69 Frac_(85) node-50 143

Frac_(23) node-19 28 Frac_(93) node-76 252

Frac_(27) node-18 43 Frac_(97) node-8 29

Frac_(30) node-18 36 Frac_(103) node-137 158

Frac_(31) node-52 79 Frac_(105) node-47 79

Frac_(38) node-176 244 Frac_(107) node-20 34

Frac_(52) node-32 50 Frac_(108) node-34 51

Frac_(70) node-90 113

9

a)

b)

Fig. 6. Scaling of radius in a) node 25 of Frac-6 hand-sample, and b) node76 of Frac-93 hand-

sample

Table 2. Radius generated from the main point 25 of Fig. 6.a

Radius

length

No. of

nodes

Radius

length

No. of

nodes

Radius

length

No. of

nodes

Radius

length

No. of

nodes

10 1 220 14 430 41 640 49

20 1 230 15 440 41 650 49

30 1 240 16 450 43 660 51

40 2 250 16 460 43 670 51

50 2 260 16 470 43 680 52

60 3 270 20 480 43 690 55

70 4 280 22 490 44 700 55

80 6 290 24 500 44 710 55

90 6 300 26 510 45 720 55

100 6 310 27 520 46 730 55

110 6 320 30 530 46 740 56

120 7 330 30 540 46 750 56

130 9 340 32 550 46 760 57

140 9 350 32 560 47 770 57

150 10 360 32 570 48 780 57

160 10 370 34 580 48 790 58

170 10 380 34 590 48 800 58

10

180 10 390 36 600 48 810 58

190 11 400 37 610 48 820 60

200 12 410 37 620 49 830 60

210 14 420 41 630 49 840 61

5 Conclusions and Future Work

In this work, a general methodology for analyzing the topology of fracture

networks is presented. In our 100 hand-samples images two groups of fractures were

identified. The first one characterizes fracture images with many nodes and short

distances between them. These characteristics may indicate larger capability for fluid

transport. The second group comprises fracture system with fewer nodes and large

distances among them making the intersections sparser than in the previous group.

We expect these results could be used as indicators in the evaluating in fluid flow

capability. They can be compared with other geophysical features associated to the

reservoir under study. This methodology can be applied to other kinds of problems

modeled as graphs where the intersection of paths is important. As a possible line of

future work, the analysis can be done from fractures obtained from RGB images,

where the first stage of this methodology will be an important part in the recovery of

the structure of the fractures.

On the other hand, it is convenient to explore other centrality metrics in complex

networks that have emerged, such as the method named page rank [24], that is used to

determine the importance of a node (used in the analysis of Web pages based on the

links pointing to them). This algorithm initiates a random walk at a random node,

following a random link at each node, with some small probability, at every step, of

jumping to a randomly chosen node without following a link. This algorithm gives

high importance (high probability of hitting) to nodes with a high number of links

pointing to them, and also to nodes pointed to by these nodes.

11

Fig. 7. Intensity of intersection points (non-terminal nodes) in different radius lengths

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