## 1. Introduction

[2] Unsaturated flow experiments in fracture networks indicate that intersections can direct flow to a single exiting fracture [*LaViolette et al.*, 2003]. In addition, they have been found to gather water from above to release as a pulse below [*Wood et al.*, 2002]. Recently we employed a simple automaton to study the consequences of these two fracture intersection behaviors embedded within a network [*Glass and LaViolette*, 2004, hereinafter referred to as I]. This “tipping bucket model” or TBM is similar to the generic directed “sand-pile” model originally studied by *Dhar and Ramaswamy* [1989] but with the added complication of stochastic, singly directed flow. The TBM idealizes the fracture network as a regular, two-dimensional array of intersections arranged on a diamond lattice (Figure 1). Periodic boundary conditions are implemented along the vertical edges of the network of 100 (horizontal) × 1000 (vertical) sites, so that water exiting on one side reappears on the other. Buckets are placed on alternate sites on the horizontal axis so that there are 50 buckets on each row. Φ is defined as the fraction of intersections, or buckets, that are connected to only one or the other but not both of the neighboring buckets in the row below. For Φ > 0, the choice of which of the two neighboring buckets to connect is random; once chosen, it remained set for the duration of the simulation. Water is added in unit increments at random to buckets along the top row and exits the network from the bottom row. Between additions, the network is relaxed by tipping the eligible buckets, as follows: when the level of water in a bucket *j* exceeds its threshold θ_{j} (which in I were all set to 10), it tips and distributes all of its volume to the (one or two) connecting buckets in the row below; the direction of the flow is always top to bottom. We obtain from operation of these simple local rules a self-organized spatial-temporal structure. For increasing Φ, channels form due to convergence within the network; spatial structure with depth transitioned from divergent to braided to the fully convergent hierarchal end member at Φ = 1. Water moves through these defining structures as pulses, or avalanches, that can penetrate to great depths. The avalanche size distribution transitions away from power-law behavior as Φ increases and convergence breaks the scaling. For only single outflow (Φ = 1), convergence is maximal and every avalanche spans the entire system but transmits the minimum volume of water.

[3] Here, we extend the TBM to consider the added realism of the dynamic overload process. Dynamic overloading occurs when a large volume of fluid is passed to a small volume intersection and causes the bucket to split its flow even if normally it would only direct the flow singly. We consider this additional process in context of a heterogeneous bucket field as is also expected in natural fracture networks. We find that as occurrence of dynamic overloading increases, the model behavior transitions from convergent flow back to divergent flow comparably to that found in I for Φ. The position of the transition is dependent on the width of the distribution and can be roughly scaled by its coefficient of variation. The divergent flow with overloading differs from that for Φ as it occurs as a natural consequence of a heterogeneous field and imposes a dynamic structure where additional pathways activate or deactivate in time.