Motivated by multiphase flow in reservoirs, we propose and study a two-species sandpile model in two dimensions. A pile of particles becomes unstable and topples if, at least one of the following two conditions is fulfilled: 1) the number of particles of one species in the pile exceeds a given threshold or 2) the total number of particles in the pile exceeds a second threshold. The latter mechanism leads to the invasion of one species through regions dominated by the other species. We studied numerically the statistics of the avalanches and identified two different regimes. For large avalanches the statistics is consistent with ordinary Bak-Tang-Weisenfeld model. Whereas, for small avalanches, we find a regime with different exponents. In particular, the fractal dimension of the external perimeter of avalanches is $D_f=1.47\pm 0.02$ and the exponent of their size distribution exponent is $\tau_s=0.95\pm 0.03$, which are significantly different from $D_f=1.25\pm 0.01$ and $\tau_s=1.26\pm 0.04$, observed for large avalanches.

中文翻译：

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入侵沙堆模型

受储层多相流的驱动，我们提出并研究了二维的两种类型的沙堆模型。如果满足以下两个条件中的至少一个，则一堆粒子变得不稳定并倾倒：1) 堆中一种粒子的数量超过给定的阈值或 2) 堆中的粒子总数超过 a第二个门槛。后一种机制导致一个物种通过另一个物种主导的区域入侵。我们对雪崩的统计数据进行了数值研究，并确定了两种不同的状态。对于大雪崩，统计数据与普通的 Bak-Tang-Weisenfeld 模型一致。而对于小雪崩，我们发现了一个具有不同指数的机制。特别地，雪崩外周的分形维数为$D_f=1.47\pm 0。