We start with the applet for Model 1 on 30×4 lattice. The sand transport in this model is directed along x-axis, i.e. an unstable sand column loses one grain of sand to its right neighbor in x-direction and its two neighbors in y-direction. Boundary conditions are periodic in y-direction and open in x-direction. Due to a directed nature of sand redistribution it is never lost at the left boundary. To indicate this fact we draw a gray stripe only at the right boundary. The description of different buttons and checkboxes on the canvas of these applets can be found here.  Repetitively adding sand at the first column to the left often causes system-wide avalanches (this way the stationary sandpile gets rid of the constant influx of sand),  but rarely changes the configuration.  To see this for yourself push the "Continue" button in the applet below and start clicking (adding sand) in the first column. You can see how avalanches roll through the lattice hardly affecting sites that are far to the right from your driving site.

Model 1:

Model 1A is a slight modification of Model 1, defined on 30×2 lattice. The sandpile dynamics is still directed, but boundary conditions in y-direction are closed, i.e. only one grain of sand goes to the nearest neighbor in the column, compared to two grains transported in model 1 with periodic boundary conditions. Now stable columns can only have heights 0 or 1.  The advantage of this model is that we understand exactly how an avalanche changes the configuration: The configuration never changes after the leftmost
column (1,1) (by this notation I mean vertical column with the first 1 under the other). In order to change this first (1,1) one needs a grain of sand added to the upper site in the configuration (1,0), (1,0), ... (1,0) all the way to the left from this stubborn (1,1).  The resulting configuration is (1,0), (1,0), ... (1,0), (0,1) and changes advance one step to the right.  Since this requires the sandpile to be in a unique state among 3^x possible SOC states, such change is exponentially unlikely for large x.

Model 1A: