Cellular Automaton Wave Propagation Models of Water Transport Phenomena in Hydrology

Robert N. Eli

West Virginia University

Hydrologists have widely accepted the proposition that water transport over and beneath the surface of the Earth can generally be modeled by partial differential equations describing the propagation of disturbance waves. Traditionally, water transport problems in hydrology have been broken up into zones of application, such as surface runoff, streamflow, unsaturated subsurface flow, and saturated groundwater flow. Although the basic conservation laws are the same for all zones, the time scales of propagation of disturbance waves vary over several orders of magnitude when comparing surface and subsurface flows. Additionally, turbulent flows dominate surface runoff and streamflow while laminar flows dominate the movement of water in the subsurface porous media. These latter differences result in disturbance wave characteristics that arise from significantly different physical processes.

In spite of these differences, it has been demonstrated that water transport within each of the hydrologic zones can be adequately represented by the so-called advection-diffusion equation. A simultaneous numerical solution to all the governing partial differential equations, for all zones, is considered computationally prohibitive. Therefore, all popular hydrologic system models incorporate solution approximations in one or more of the zones that effectively breaks the continuous propagation of disturbance waves that are believed to be important in describing the transport of water within a typical watershed.

From a conceptual viewpoint, cellular automata (CA) would appear to provide a possible solution to the wave propagation modeling problem. Examples of cell property advection and diffusion processes appear in numerous examples in NKS, and the CA literature. In beginning the CA development process, simple one-dimensional CA rules have been created that represent the advection of mass (water), with the appropriate diffusive behavior. Each cell contains integer numbers of discrete particles that can be passed to its neighbors. The number of particles passed in each time step is dependent on the number of particles in the adjacent neighboring cells and the cell “properties”. Cell properties consist of a “conductance” and a “storativity”. Evolution of the CA using the rule demonstrates good agreement with existing analytical solutions of the advection-diffusion equation in the surface runoff zone and the saturated groundwater flow zone.