Cellular Automaton Wave Propagation Models

1
Cellular Automaton Wave Propagation Models of
Water Transport Phenomena in Hydrology Robert N.
Eli Associate Professor Department of Civil and
Environmental Engineering, West Virginia
University Hydrology is the study of the
occurrence and movement of water in natural
systems. Hydrologists have observed that the
motion of water over and beneath the earths
surface can be adequately described by the
physical processes of advection (mass movement)
and diffusion (attenuation) in the vast majority
of situations (Ponce 1989, Wurbs James 2002,
Abbott Basco 1989, Bear Verruijt 1987). This
means that wave propagation can be treated as a
advection-diffusion process rather than a fully
dynamic process (in other words, momentum effects
can safely be ignored). The governing
differential equation for advection-diffusion
is x is the space dimension and t is time, and
h is either the depth of the flowing water on the
surface, or the depth of the saturated soil in
ground water flow. The velocity c and the
diffusion coefficient D dictate the advection and
diffusion characteristics of the property h in
the particular hydrologic process (either surface
water flow or ground water flow). i is the rate
of depth increase due to either rainfall or
infiltration. Surface Water Flow An example of
surface water flow is that produced by rainfall
on a parking lot. In this example the advective
velocity c and the diffusion coefficient D are
given by (Ponce 1989) S is the slope of the
surface and n is Mannings n (a surface roughness
coefficient (Ponce 1989)). Ground Water Flow An
example of ground water flow is that produced by
the movement of water in a layer of porous
granular material, such as sand, bounded on the
lower side by a horizontal impervious surface. In
this example the advective velocity c and the
diffusion coefficient D is given by (Bear
Verruijt 1987) KH is the hydraulic
conductivity of the sand (ability of the sand to
let water flow without resistance), CS is the
storage coefficient of the sand (fraction of the
total sand volume that is available to hold
water), and ?h/?x is the rate of change of h in
the x direction (slope of h). Solution
Methods Equation 1 can be solved in
application to practical hydrology problems (such
as the examples above) by a range of numerical
methods (computer-based algorithms). Finite
difference methods are the most widely used and
they are divided into two categories, explicit
and implicit. Implicit methods are more difficult
to program and require more computing resources
in the form of calculation time and information
storage. Explicit methods are easier to program
and require less computing resources.
Additionally, Cellular Automata (CA) and the
explicit method (EM) share similar
characteristics, hence a classic explicit method
is compared to a new cell based method in this
proof of concept study. Explicit Finite
Difference (QUICKEST) method The space
dimension x is divided into a row of cells, each
having a width of ?x, starting at time t. If the
cell location in the x direction is indexed with
j, then the depth h at the next time step t ?t
(the time increment is ?t) is a function of cells
j-2, j-1, j, and j1 at time t. The QUICKEST
method (Abbott Basco 1989) algorithm is 3rd
order accurate (good accuracy) and is applied to
each cell in the x direction Conservative
Mass Cellular Automata (CMCA) method A serious
concern with most finite difference methods is
that they do not conserve mass, causing mass to
appear to be created or destroyed as the
computations proceed in time. The CMCA method
avoids this problem by explicitly exchanging mass
(in this case, water volume) between the cells
during each time step. The advective water volume
V 7 leaving each cell is sent to an adjacent
cell, depending on the direction of uocomputed in
2 or 4. The diffusive volume v 8 leaving
each cell is sent to the adjacent cells by
apportioning v to both the left and right cell as
a function of the magnitude of the down gradient
?h/?x to the left or right. If there is no down
gradient in h to either the left or right, or
both, no portion of v is sent to those adjacent
cells. In 8, ? is given by 6 above, while
CS 1 for surface water flow, or is less than 1
for groundwater flow (see ground water paragraph
above). References Abbott, M.B. and Basco, D.R.,
1989 Computational Fluid Dynamics, An
Introduction for Engineers Longman Scientific
Technical, John Wiley Sons, Inc., New York,
NY. Bear, A.V. and Verruijt, A., 1987 Modeling
Groundwater Flow and Pollution, Theory and
Applications of Transport in Porous Media D.
Reidel Publishing Co. Boston, MA. Ponce, Victor
M., 1989 Engineering Hydrology, Principles and
Practices Prentice Hall, Englewood Cliffs,
NJ. Wurbs, R.A. and James, W., 2002 Water
Resources Engineering Prentice Hall, Upper
Saddle River, NJ.

Surface Water Flow The Figures in this box
compare the results of running the Quickest and
CMCA methods. The impervious surface is 200 meter
long and 1 meter wide with a slope of 0.001 (1 m
per 1000 m). Rainfall occurs over the entire
length of the surface at a rate of 0.072 m/hr,
beginning at time 0.0 and ending after 5 hours.
The length is divided into 20 cells (?x 10 m),
and ?t 60 sec . The duration of the simulation
is 10 hours. The surface is initially dry, and
the water layer builds up on the surface over
time and flows off the end of the surface at the
200 m end of the scale.
Figure 3 A space-time diagram of the depth of
water on the plane surface (using the CMCA
method).
Figure 1 This shows how the water builds up in
depth over the length of the surface for the
first 3 hours of the rainfall, plotted at equal
intervals of time. The CMCA method is used.
Figure 4 A space-time diagram of the diffusive
volume transfer component of the CMCA simulation.
It shows that the maximum diffusion is occurring
during water depth buildup at the downstream
boundary condition (a problem ?!)
Figure 2 This shows a comparison of the flow
rate (discharge) leaving the end of the plane
surface for the two computational methods. The
conservation of mass error for each method is
Quickest -10.62 CMCA 0.00
Ground Water Flow The Figures in this box
compare the results of running the Quickest and
CMCA methods. A 200 meter long and 1 meter wide
horizontal impervious surface is topped with a
thick layer of sand. The length is divided into
20 cells (?x 10 m), and ?t 300 sec . Water
infiltrates at a rate of 0.036 m/hr into the sand
in a zone extending from 45 m to 145 m. The
infiltration continues for a period of 24 hours.
The boundary conditions are equivalent to a
vertical impervious barrier at 0 distance and
exposed outflow at the 200 m end of the scale.
The simulation begins with the sand being totally
dry and the duration of the simulation is 60 days.
Elapsed time, days
Figure 3 A space-time diagram of the depth of
saturated sand (using the CMCA method).
Figure 5 This shows the buildup in the thickness
of the saturated sand at equal intervals of time
over the first 16 hours of the simulation using
the CMCA method.
Elapsed time, days
Elapsed time, days
Figure 4 A space-time diagram of the diffusive
volume transfer component of the CMCA simulation.
It shows that the maximum diffusion is occurring
during the infiltration period at the beginning
of the simulation, when water table slope changes
are at their maximum (expected).
Figure 6 A comparison of the flow rate
(discharge) exiting from the sand at the 200 m
end of the scale over the duration of the
simulation. The conservation of mass error for
each method is Quickest 6. 01 CMCA
0.90