Fuzzy graph theory appl ied to brittle plane network-A need for carbon sequestration models

Improved carbon sequestration (CCS) models with rocks as sinks require incorporation of uncertainty into the models. In such cases of uncertain geoscientific problems, fuzzy graph theory can be useful. Brittle shear plane network with indistinct shear planes is common in natural sheared rocks, and can be targeted for CCS. Due to non-unique possibility of continuity of P-planes, it is not possible to represent such networks as crisp graphs. We present few natural examples of the former type of P-planes in shear zones, and how fuzzy graph theory can represent the fracture network and fluid flow. The process involves assigning some sample numerical probability to represent the connectedness between the underdeveloped Pplanes and the Y-planes. The presentation is a geometric exercise and does not extend to the genesis of the shear zones.


Introduction
Structural geological modeling requires representation of structures into some numerical form.
When the structures are clearly decipherable, this is rather easy, for example, one can apply the graph theory (e.g., Sanderson et al., 2018;Mukherjee 2019). Classical graph theory works when there is a full certainty that which nodes are connected by which vertices through edges.
Various brittle plane geometries develop in rocks at shallow crustal depths that undergo brittle shear (Fig. 1a). In case the shear planes are clearly delineated (Fig. 1b), one can use graph theory to represent the brittle plane o f sigmoid shear bound by parallel planar shear planes are commonly seen in sheared rocks.   Mukherjee (2019). Given that its nodes are a, b, c, d, e and f, the adjacency matrix is: Carbon sequestration (CCS) has recently been modeled in terms of extraneous CO2 sources and CO2 sinks in rocks and P-graph modeling approach has been undertaken (Chong et al. 2014 (Misra and Mukherjee 2018), and yet the modeler needs to make some presentation of the brittle plane network. In such cases, where the classical graph theory cannot represent the brittle plane network, fuzzy graph theory should be used.
The concept of fuzzy sets was introduced by Zadeh (1965), which later was well applied to solve several research problems that are uncertain in nature. Crisp set is a well-defined collection of distinct objects. If there is any vagueness in the description of objects, the concept of fuzzy set is used. In other words, a fuzzy set is used to represent qualitative data. The crisp set is not able to work with scientific problems with inherent uncertainty because it consists of just two truth values: 0-false and 1-true. These single values of membership degree are unable to manage the uncertainties. Atanassov (1999) proposed the intuitionistic fuzzy set to manage uncertain situations using an extra degree of membership, defined as the hesitation margin.
Intuitionistic fuzzy set is as an extension of Zadeh's theory of fuzzy sets. Compared to classical fuzzy set it is more flexible and efficient to work with uncertainty due to the presence of hesitation margin.

Example
Figs. 2a-c present a network of brittle shear planes where not all P-planes can be tracked with confidence. Here the curved P-plane AB joins two sub-parallel Y-planes.
However, the other P-planes (CD, KZ, EF, GH and IJ) and do not join both the Cplanes. In some cases the P-planes are too close-spaced in order to distinguish them confidently in naked eyes. A simplified situation can be thought in this case (Fig. 3).  Fig. 3. An idealized brittle shear plane network. Few sigmoid P-planes (e.g., AB) join the C-planes. The other P-plane, CD, KZ, EF, GH and IJ, are not fully developed and merge with just a single C-plane. In such a situation, fuzzy graph theory has been applied in this work.
We represent fracture network using intuitionistic fuzzy graph as follows (Fig. 4). Points We will now get into further detail of fuzzy graph theory to tackle flow problems. The adjacency matrix of a fuzzy graph G: (V,σ,µ) is an n×n matrix defined as A= [aij] where aij=µ(vi,vj) (Anjali and Mathew 2013). Fig. 3. In this fuzzy graph, the numbers inside the first bracket represents the node weights, which do not appear in the adjacency matrix since. Instead only the arc weights are used for the entries in the adjacency matrix. Such a presentation is as per Anjali and Mathew (2013).

Fig. 4. A fuzzy graph corresponding to the case of
The entries in the adjacency matrix are obtained as follows. Since v1 is not shown to be connected with itself by any edge, the entry is 0. Since v1 is adjacent with v2 with arc weight 0.1, the entry is 0.1 and so on.
Consider the network in Fig. 6. From v2 to v1 the flow can vary in magnitude from 7 to 2 units. Therefore we take fuzzy weights 0.7 for v2 and 0.2 for v1. Similarly, one can take the fuzzy weight of v3 as 0.3, v4 as 0.1. This is a first-time approach through this work. Suppose the fuzzy graph presented in Fig 6 has no arrows on edges. Then its corresponding adjacency matrix will differ. That means flow can happen from one vertex to another and also vice versa. The definition of a fuzzy graph allows us to give the arc weights in the following way. The arc (edge) weights are taken as the minimum among the corresponding node weights. This is logically true, since the maximum flow between two nodes is the minimum capacity among the two nodes. In that case, the adjacency matrix is: Suppose, instead of a range of flow, definite units of flow were given (e.g., 7, 5 and 2). We then assign fuzzy weights as 0.7, 0.5 and 0.2, respectively, for arcs. If a range of flow is given, we will get a freedom to take the minimum value amongst the range for representing the arc weight for the range of flow. As per the definition of adjacency matrix, one can represent these values by using their corresponding fuzzy weights. If 2, 1 and 3 are avoided, i.e., flow range 7 to 2 becomes 7, flow range 2 to 1 becomes 2 and flow range 5 to 3 becomes 5, the flow becomes exact, and one can assign the fuzzy weights. In fuzzy graph, we can give the maximum value of node weight and arc weight as 1 and minimum as 0. All other values are in between 1 and 0. A value equal to 1 means that the flow has fully happened. The intensity of the flow is represented by using these fuzzy weights. Ramakrishnan and Lakshmi (2008) discussed how to fuse two nodes of the same fuzzy graph as follows. Let G: (V, σ, µ) be a fuzzy graph and let , ∈ . By the join (fusion) of two vertices u and v, the following is meant.    Fig. 9. Fusion of two vertices u and v from two fuzzy graphs 1 and 2, , respectively into . The fused vertex is assigned a weight 0.9, which is the maximum weight in between and .
The resulting fuzzy graph is Guv= (σuv,µuv) where: Coming to the case of a directed fuzzy graph or a fuzzy digraph, consider the network in Fig 10. The adjacency matrix of this new fuzzy graph is given below.
In the BB1 fused vertex, 0.3 is stated inside bracket. This means that the maximum out of B (0.3) and B1 (0.1) to be stated: "BB1 (0.3)". For the maximum taken, the logic is that the capacity elevates when nodes are fused (Ramakrishnan and Lakshmi 2008).

Discussions and Conclusions
Indistinct or poorly developed brittle plane networks are common in rocks. Their numerical representation can be important for modelers, such as in fluid flow, CCS and hydrocarbon reservoir studies. As the hydrocarbon resources are depleting, flow models realistic to geologic cases are of paramount importance. In this article we introduce the concept of fuzzy graph theory as a first step to fulfill such a far-reaching aim. An example of brittle shear planes consisting of Y and P-planes was considered with different degrees of connection between them. We further presented theoretical issues regarding fuzzy digraphs and joining/fusion of vertices.