Chaotic
Models of Dripping Water From a Fracture Under Ponded Infiltration
at Hell's Half Acre, Idaho
Boris
Faybishenko
Contact: Boris Faybishenko,
510/486-4852
bfayb@lbl.gov
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the PDF File Here]
Research
Objective
Preliminary
analysis of the results of infiltration tests from the Hells
Half Acre (HHA) field site, Idaho, showed that under some conditions,
flow in fractured rocks can be described using chaotic models (Podgorney
et al., 2000). The objective of this project is to perform a nonlinear
dynamical analysis of the time variations of infiltration and outflow
rates and dripping water phenomena observed during the HHA tests
in order to determine phenomenological models describing spatial
and temporal chaotic behavior of flow in fractured rocks.
Approach
The
results of three small-scale ponded infiltration tests conducted
in fractured basalt during the summer of 1998 at the Hells
Half Acre field site were analyzed. The tests were conducted using
a small reservoir (40 x 80 cm) constructed on the surface exposure
of a fracture intersecting an overhanging basalt ledge (Podgorney
et al., 2000). The spatial and temporal behavior of inflow and outflow
rates, including temporal and spatial water dripping from the undersurface
of the ledge were monitored. The data analysis was conducted using
the phase-space reconstruction of one-dimensional time-series of
water dripping intervals at different locations. The main idea behind
the reconstruction of the system dynamics from one-dimensional scalar
data is the evaluation of diagnostic parameters of chaos, such as
the correlation time (Dt),
global embedding dimension (GED), local embedding dimension (LED),
Lyapunov dimension (LD), Lyapunov exponents (LE) and correlation
dimension.
Accomplishments
The
results of this study show that a dripping water behavior is transient
and either quasi-periodic or nonperiodic. The dripping-water behavior
occurs on three temporal scales, such as seconds, hours and days,
which are not related to changes in boundary conditions. The observed
nonlinear behavior is caused by a superposition of several physical
nonlinear processes generating chaos for flow in unsaturated fractured
rocks, including the capillary barrier effect. The observed variations
of the flow rate and dripping intervals are apparently caused by
a combination of both deterministic-chaotic, reflecting the physical
deterministic nature of nonlinear flow and transport processes,
and random components (Faybishenko, 1999). The volumetric outflow
rates combined from several dripping locations exhibit spatial and
temporal instabilities with primary low-frequency fluctuations and
secondary high-frequency fluctuations caused by local instabilities.
The time series data and corresponding attractors indicated several
routes to chaos in water dripping processes, such as intermittency
fluctuations, bifurcation, gradual and/or rapid collapse of stability.
It was determined that different models, such as deterministic,
deterministic-chaotic, stochastic-chaotic and random models, can
be used to describe the data for different times. Figure 1 illustrates
an example of a time-series of water-dripping intervals exhibiting
deterministic chaos and a 3-D phase-space attractor.
Significance
of Findings
If
a flow system exhibits a deterministic chaotic behavior, its long-term
predictability is limited. For such a system, one can provide precise
short-term predictions and only a range of possible long-term predictions.
The models developed in this project would be of interest for investigations
of dripping phenomena in fractured rocks at several DOE sitese.g.,
at the potential nuclear waste repository at Yucca Mountain, Nevada,
and in fractured karst at Oak Ridge, Tenn.
Related
Publications
Faybishenko,
B., Evidence of chaotic behavior in flow through fractured rocks,
and how we might use chaos theory in fractured rock hydrogeology,
Proc. Dynamics of Fluids in Fractured Rocks: Concepts and Recent
Advances, pp. 207-212, Berkeley Lab report LBNL-42718, 2000.
Podgorney,
R., T. Wood, B. Faybishenko and T. Stoops, Spatial and temporal
instabilities in water flow through variably saturated fractured
basalt on a one-meter scale, AGU Monograph 122, Dynamics of Fluids
in Fractured Rocks, in press.
Acknowledgements
This
work has been supported by the Environmental Management Science
Program of the U.S. Department of Energy under Contract No. DE-AC03-76SF00098
and by Idaho National Engineering and Environmental Laboratory.
The field infiltration tests were conducted by R. Podgorney and
T. Wood of INEEL.
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Figure
1. An example of a 3-D attractor plotted for 2,078 water dripping
intervals (n) with Dt=1,
showing a determinsitic structure with a secondary noisy component.
Parameters of chaos are: GED = 5, LED = 5, LD = 4.611, largest
LE = 0.364, minimum LE = -0.724. (Test 8, Point 6, 7/6/98).
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