A discrete time evolution model for fracture networks

We examine geophysical crack patterns using the mean field theory of convex mosaics. We assign the pair $(\bar n^*,\bar v^*)$ of average corner degrees to each crack pattern and we define two local, random evolutionary steps $R_0$ and $R_1$, corresponding to secondary fracture and rearrangement of cracks, respectively. Random sequences of these steps result in trajectories on the $(\bar n^*,\bar v^*)$ plane. We prove the existence of limit points for several types of trajectories. Also, we prove that cell density $\rho = \bar v^*/\bar n^*$ increases monotonically under any admissible trajectory.


Introduction
Fragmentation is one of the most ubiquitous natural processes and the efforts to decode its geometry have been at the forefront of geophysical research [1][2][3][4][5][6][7][8].While many aspects of fragmentation are inherently three dimensional, the 2D aspects of the phenomenon are interesting in their own right: the most visible fingerprints of fragmentation are surface fracture patterns (also called 2D fracture networks) on various scales [4], ranging from mud cracks resulting from desiccation [6,7,8] (see Figure 1 for two examples) through basalt columns [3] to the pattern defined by the tectonic plates [2,9].In [2] the geometric theory of tilings [13,14], in particular, the mean field theory of convex mosaics [10,12] was applied to identify fracture networks with a point on the so-called symbolic plane, spanned by two characteristic geometric features, the nodal and cell degrees [10,14]   However, fracture networks are not static objects; they evolve in various manners.
Evolution models are popular mathematical tools, used also in operations research [16].This particular evolution may be modeled by regarding an initial fracture network (to which we will refer as primary network) and then consider a family of discrete, local events under the random sequence of which the primary network evolves.While these events have been described in [2], the evolution model has not been developed and this is the goal of the current paper.Such a model would, for example, admit the following In the model we will consider two types of local events which evolve the fracture network.One such local event is undoubtedly secondary fracture where an existing fragment particle (produced in primary fracture) is being split into two parts along a (random) fracture line; this can be observed on rock outcrops.Another local event is when, in the process of crack healing and rearrangement "T" nodes evolve into "Y" nodes; this can be observed in drying mud and also in columnar joints [6,8].
Our evolution model will only consider these two steps.Needless to say, the model could be refined by considering other local events.However, even this simple model suffices to show the principles of how such models operate in general and, as we will show, the model based on these two steps has already rich dynamics and promises to explain several features of the observed natural processes.
Our paper is structured as follows: in Section 2 we introduce all necessary mathematical concepts.Section 3 is dedicated to the development of the model and the main results and in Section 4 we draw conclusions.

Mathematical concepts
As models of 2D fracture networks we consider convex, normal tilings which we define below, following [11,12]: Normal tilings are 2 dimensional tessellations where each cell is a topological disk, the intersection of each of the two cells is either a connected set, or the empty set and the cells are uniformly bounded from below and above.

Definition 2.
Convex tilings are 2 dimensional tessellations where each cell is convex.

Remark 1.
As a direct consequence of Definitions 1 and 2, we see immediately that all cells of normal, convex tilings of the Euclidean plane are finite, convex polygons (see also [11].)We can also see that normal tilings of infinte domains (such as the Euclidean plane) have infinitely many cells while normal tilings of finite (compact) domains have a finite number of cells.

Definition 3.
A node of a convex, normal tiling is a point where at least 3 cells overlap.

Remark 2.
As a consequence of Remark 1 and Definition 3, we can see that in a convex, normal tiling the boundary of each cell contains at least 3 nodes.

Definition 4.
In a normal, convex tiling the combinatorial degree v of a cell is equal the number of nodes on its boundary and the combinatorial degree n of a node is equal to the number of cells overlapping at that node.
Following [15], we also introduce a concept, which is more sensitive to the actual shape of the cells: In a normal, convex tiling the corner degree v* of a cell is equal to the number of its vertices and the corner degree n* of a node is equal to the number of vertices overlapping at that node.

Remark 3.
We can immediately see that for all cells and nodes we have  * ≤ ,  * ≤  .

Definition 6.
We call a node regular if  * = .We call a normal, convex tiling regular is all nodes are regular.We denote the number of all nodes by V and the number of irregular nodes by VI and the quantity  = −   * is called the regularity of the tiling,  = 1 corresponding to regular tilings.(We remark that for regular tilings we have  * =  for all cells.We also see that for regular nodes we have  * ≥ 3 whereas for irregular nodes we have  * ≥ 2.) The quantities ,  * ,  * can be easily averaged on finite mosaics: Let us denote the number of faces (cells), edges and vertices (nodes) of a finite portion of a convex, balanced tiling by F,E,V, respectively.We call the quantities .We will refer to , , ,  * as the fundamental quantities of the tiling.

Remark 4.
Henceforth we only consider normal, convex tilings and for all infinite tilings we require that they are balanced.

Remark 5.
Based on [11,12], for infinite, balanced, convex (a) regular and (b) irregular tilings of the Euclidean plane we have: ( For infinite tilings M we use the limit process described in Definition 6.

Remark 6.
We can immediately see that () .So, the cell density can be expressed by the average cell degree and average nodal degree, but its value is independent of the definition of these degrees.We can also see that ̅ () =  lines correspond to rays passing through the origin of the symbolic plane.

An evolution model for fracture networks
Our model is inspired by [2], where planar fracture patterns have been analyzed and classified in the symbolic plane, based on their nodal and cell averages.Here we take one further step and build a model to study not only their current state but also their evolution in the symbolic plane.
We start by defining our main hypothesis.While we believe that this hypothesis captures several aspects of physical fragmentation, our goal is not just to define this particular evolution model but also to demonstrate how such a model is constructed and how it operates.

Hypothesis 1.
A. The fracture network is a finite domain of a convex, balanced tiling of the Euclidean plane.
B. Evolution of the crack pattern takes place in a series of discrete events, consisting of 2 step-types (R0,R1) to be detailed below.The order of the step-types is important: we assume that the first k pieces of R1 steps are preceded by at least (k/2) pieces of R0 steps.
(This is explained in detail below when we define step R1.) C. (1) During secondary cracking, one cell of the primary crack network is split into two parts along a straight line segment, connecting two points belonging to the relative interior of two different edges of the cell.(R0 type step).It is easy to see that R0 type steps retain the convexity of the initial mosaic.See Figure 2.
(2) During crack healing-rearrangement, the edges and nodes of the crack network are rearranged so that "T" nodes evolve into "Y" nodes [4] and each such event corresponds to an R1-type step.Obviously, this step can only be performed on an irregular "T" node.
Since we assumed the initial mosaic to be regular and one step of type R0 generates two irregular nodes, the first k pieces of R1 steps must be preceded by at least (k /2) pieces of R0 steps.See Figure 2. Observe that the first k pieces of R1 steps must be preceded by at least (k /2) pieces of R0 steps.
For clarity, below we summarize in Table 1 how the steps R0 and R1 operate on the fundamental quantities , , ,  * of the tiling.See also Figure 2  and let us denote by ( ̅  * , ̅  * ) the limit point on the symbolic plane, reached after applying an infinite random sequence of R0 and R1 steps, with respective probabilities 1-p, p and we call this a -trajectory.

Remark 7:
We mention two special -trajectories: 0-trajectories correspond to sequences consisting entirely of R0-type steps and 1-trajectories correspond to sequences consisting entirely of R1type steps.We also mention that if the mosaics corresponding to any finite number of evolution steps R0 and R1 are, like the initial mosaic , finite, convex, normal tilings.However, the limit point ( ̅  * , ̅  * ) does not correspond to a normal mosaic, it should be regarded as a point of the symbolic plane.
The following lemma applies to general -trajectories: Let M be a finite, normal, convex mosaic characterized by an initial state ( ̅  * , ̅  * ).Then, for all -trajectories we have Proof : (a) After the first c(1-p) steps of type R0 we have: Q.e.d.
The result of Lemma 1 is illustrated in Figure 3 where we computed standard -trajectories for

Remark 8:
We can see that all limit points lie on the ̅ * = 2 ̅ * line.However, the physically relevant portion of the trajectory may terminate earlier.This is easiest understood if we consider the following argument: when constructing physically relevant p-trajectories we also have to consider Hypothesis 1, C(2) which implies that the necessary condition for an infinite ptrajectory to be physical is  ≤ 2/3, as at least half as many R0 steps are needed as we have R1 steps.For  ≥ 2/3, the physical part of the trajectory will terminate as the mosaic becomes regular at (or, for finite mosaics, near) the hyperbola defined in (5)(a).Beyond this point, the trajectory exists only in an algebraic sense (as a sequence of numbers) to which we do not attach any direct geometric interpretation.We also mention that, as  → 1, the tangent of the trajectory at ( ̅  * , ̅  * ) will approach the ̅ =  line with constant cell density passing through Needless to say, -trajectories represent only a rather restriced class of geophysical evolution processes for fracture networks.In such a process, the ratio of R0-type and R1-type evolution steps remains constant.However, our model also admits statements of more general type.
Relying on definitions 10 and 11 we make the following observation:

Lemma 2:
In the discrete-time fracture network evolution model formulated in Hypothesis 1, the cell density ̅ () increases monotonically over time.

Proof:
Since there are two steps in the model (R0 and R1), so, if we can show separately for both step-types that the cell density does not decrease, we have confirmed the statement of Lemma 2.
As described above, in the crack network evolution model formulated in Hypothesis 1, the cell density is either constant or increasing in each step.Q.e.d.
So far, we did not make any additional restrictions on the initial mosaic M beyond requiring that it should be a convex, normal tiling.However, typical primary fracture networks have special position on the symbolic plane; in [2] it was argued that if the network is created by long straight fracture lines, then this corresponds to the point ( ̅ * , ̅ * ) =(4,4) of the symbolic plane.Motivated by this geophyiscal observation we formulate

Lemma 3:
In the discrete-time fracture network evolution model formulated in Hypothesis 1, if we choose an initial (starting) mosaic with ̅  * () < 4 then the cell corner degree ̅ * increases strictly monotonically over time.
the combinatorial degrees of the tiling, where  = ∑    =1 = ∑    =1 and   ,   denote the combinatorial degrees of the  th node and cell, respectively.Similarly, are called the corner degrees of the tiling, where  * = ∑   *  =1 = ∑   *  =1 no infinite tilings for ̅ > 2 ̅  ̅−2 and no convex tilings for ̅ > 2 ̅.Definition 10.Let M be a finite part of a regular, normal, convex tiling and let M have F faces (cells) and V vertices (nodes).Then we call ̅ () =   the cell density of M.

Figure 2 :
Figure 2: Geometry of evolution steps R0 and R1 used in the model based on Hypothesis 1.

Table 1 :
for illustration.Evolutionary equations for the steps R0 and R1 used in the model based on Hypothesis 1.