GEOPHYSICAL RESEARCH LETTERS, VOL. 40, 1–5, doi:10.1029/2012GL053885, 2013
Subsurface mass transport affects the radioxenon signatures that are
used to identify clandestine nuclear tests
J. D. Lowrey,1 S. R. Biegalski,1 A. G. Osborne,1 and M. R. Deinert1
Received 14 September 2012; revised 19 November 2012; accepted 20 November 2012.
distinguished from those of other sources and the ratios of
131m
Xe, 133mXe, 133Xe, and 135Xe can be used to help do this.
[3] When a fission weapon detonates, it produces radioxenon directly as well as precursors that decay to xenon. What
comes out of the ground is a mixture of xenon isotopes from
both of these sources. Figure 1 shows the activity ratios for
133m
Xe/131mXe versus 135Xe/133Xe that would result from a
235
U fueled weapons test in which radioxenon precursors
are absent or present [Kalinowski et al., 2010]. Here “fully
fractioned” corresponds to the xenon signal that would be
expected if all the isotopes that decay to xenon are lost and
do not contribute to the signal. The “nonfractioned” curve
refers to the xenon signal that would be expected if all the
isotopes that decay to xenon are contained and contribute
to the signal. Figure 1 also shows the radioxenon signature
that would result from the operation of a commercial power
reactor [Carman et al., 2002; Kalinowski et al., 2010; Le
Petit et al., 2008]. It is currently assumed that the relative
abundance of anthropogenic xenon isotopes is solely a function of how and when they were produced [Le Petit et al.,
2008]. As a result, it has been suggested that a radioxenon
signal that falls between the nonfractioned (red) and fully
fractioned (blue) curves can be assumed to determine
whether or not a nuclear weapon has been detonated [e.g.,
Kalinowski et al., 2010; Saey, 2009].
[4] Work done with nonradioactive tracers at the Nevada
Test Site has shown that heavy gases released below
ground will diffuse to cracks in the geology along which
they then preferentially move to the surface. The rate of
movement within cracks is itself strongly affected by variations in barometric conditions, being largest during periods
of decreasing pressure [Carrigan et al., 1996, 1997]. The
effect of fluctuations in atmospheric pressure on the transport of nonradioactive gasses through dry media has been
modeled using a double porosity model for the medium
through which the gas travels. Here a convection-diffusion
formulation is applied along with the assumption that the
geology is comprised of homogenous slabs of material
through which vertical cracks run—a reasonable assumption for many locations [Carrigan et al., 1996, 1997; Chen,
1989; Gringarten, 1984; Neretnieks and Rasmuson, 1984;
Nilson and Lie, 1990; Nilson et al., 1991].
[5] We have extended the double porosity approach to
include the effect of radioactive decay in the transport equations. Previous work with this formulation suggests that
some forms of geology combine to affect xenon isotope
ratios that result from well-contained underground tests
[Lowrey et al., 2012]. Here we show that the region between
the fully fractioned and nonfractioned curves is in fact too
narrowly defined to encompass the xenon signals that could
result from a nuclear test. The xenon signature of a
[1] The ratios of noble gas radioisotopes can provide critical
information with which to verify that a belowground nuclear
test has taken place. The relative abundance of anthropogenic
isotopes is typically assumed to rely solely on their fission
yield and decay rate. The xenon signature of a nuclear test is
then bounded by the signal from directly produced fission
xenon, and by the signal that would come from the addition
of xenon from iodine precursors. Here we show that this
signal range is too narrowly defined. Transport simulations
were done to span the range of geological conditions within
the Nevada Test Site. The simulations assume a 1 kt test and
the barometric history following the nuclear test at Pahute
Mesa in March 1992. Predicted xenon ratios fall outside of
the typically assumed range 20% of the time and situations
can arise where the ground level signal comes entirely from
the decay of iodine precursors. Citation: Lowrey, J. D., S. R.
Biegalski, A. G. Osborne, and M. R. Deinert (2013), Subsurface mass
transport affects the radioxenon signatures that are used to identify
clandestine nuclear tests, Geophys. Res. Lett., 40, doi:10.1029/
2012GL053885.
1. Introduction
[2] On 3 October 2006, the Democratic People’s Republic
of Korea gave warning of its intention to conduct a nuclear
test, and 6 days later claimed that one had been successfully
carried out. Radioxenon isotopes were the only fission products to be measured off-site afterward and served as critical
evidence that a nuclear explosion had taken place [Ringbom
et al., 2009]. Anthropogenic isotopes are in fact the only definitive evidence for nuclear test and are an important component
of a broader verification system [Hannon, 1985; Zuckerman,
1996]. Because of their short half-lives, relatively high production yields, and ability to move through geological structures,
radioxenon isotopes are ideal for verifying that a nuclear
explosion has taken place [Bowyer et al., 2002, 2011; Carman
et al., 2002; Saey, 2009; Van der Stricht and Janssens, 2001].
However, several xenon isotopes are also produced by other
anthropogenic sources such as commercial nuclear reactors
and medical isotope production [Biegalski et al., 2010;
Bowyer et al., 2011; Kalinowski et al., 2010]. As a result,
it is essential that the radioxenon signatures of a weapon be
1
Department of Mechanical Engineering, The University of Texas at
Austin, Austin, Texas, USA.
Corresponding author: M. R. Deinert, Department of Mechanical Engineering, The University of Texas at Austin, 1 University Station, C2200,
Austin, TX 78715 USA. ([email protected])
©2013. American Geophysical Union. All Rights Reserved.
0094-8276/13/2012GL053885
1
LOWREY ET AL.: SUBSURFACE MASS TRANSPORT AFFECTS RADIOXENON SIGNATURES
Figure 1. Xenon isotope ratios for different sources. The plot shows expected radioxenon signals generated from three
types of sources. Here “fully fractioned” corresponds to the xenon signal that would be expected from an open air nuclear
detonation, where all of the isotopes that decay to xenon are lost and do not contribute to the signal. The “nonfractioned”
curve refers to the xenon signal expected from a belowground nuclear test, where all the isotopes that decay to xenon are
fully contained and thus contribute to the signal. The signature generated by a commercial power reactor is also shown.
The region between the nonfractioned (red) and fully fractioned (blue) curves is currently assumed to indicate that a
nuclear weapon has been detonated. The data were generated by performing a depletion calculation using ORIGEN 2.6.
pressure at time t. Here a semicolon indicates that the symbol to its right is held constant
belowground test is more appropriately bounded by the fully
fractioned curve and the signal that would come solely from
the decay of iodine precursors.
@pðx; yÞ
@
@pðx; yÞ
¼
am
@t
@x
@x
Z d
@pð0; yÞ fm 2 @pðx; yÞ
@
@pð0; yÞ
¼
dx þ
af
@t
@t
@y
@y
ff 0
2. Methods
[6] The position-dependent concentration of the i’th isotope is given by [Chen, 1989; Lowrey et al., 2012; Nilson
and Lie, 1990]
fm
@C ðx; yÞi @ @
@C ðx; yÞi
C ðx; yÞi vðx; yÞ ¼
f m Di
þ
@x
@x
@t
@x
(4)
[9] The notation p(0,y) indicates that equation (4) represents the rate of change of pressure in the fracture, where
the coordinate x is equal to zero. The pneumatic diffusivities
of the matrix and fracture, am and af, {m2/s}, are assumed to
be constant at a given depth.
[10] The flow velocities, u(x,y) and v(x,y), in equations (1)
and (2) are functions of the differential pressures within the
cracks that arise from variations in atmospheric pressure.
The diffusivity of each species is taken to be mass-dependent [Carrigan et al., 1996] and a function of the matrix tortuosity [Chen, 1989; Gringarten, 1984; Neretnieks and Rasmuson, 1984]. Details on how Di, u(x,y), v(x,y), am, and af,
are computed are given in the auxiliary material.1
[11] Equations (1)–(4) are discretized by using first-order
backward differencing for first-order derivatives and secondorder centered differencing for the second-order diffusive
operators. The discretized equations are formulated as a set
of tridiagonal matrix equations. Solution of these equations
for each time step is made by Gaussian elimination with
periodic boundary conditions at the interior of the matrix
and a closed bottom boundary. It is assumed that there is
no interaction between vertical layers in the matrix, and
equations (1)–(4) can therefore be solved separately for each
layer, and for the fracture.
[12] Equation (1) is solved at every layer to find the isotopic concentrations within the matrix. The concentrations are
(1)
fm li C ðx; yÞi
@C ðx; yÞi @ C ðx; yÞi uðx; yÞ ¼
þ
@y
@t
Zd=2
@C ðx; yÞi
@
@C ðx; yÞi
f f Di
fm
dx þ
ff li C ðx; yÞi
@y
@t
@y
(3)
ff
(2)
0
[7] Here fm and ff are the matrix and fracture porosities
{dimensionless}, C(x,y)i is the concentration of the i’th isotope {Ci/m3}, v(x,y) is the bulk flow velocity {m/s} through
the matrix, Di {m2/s} is the diffusion coefficient of the i’th
xenon isotope, li is its decay constant {1/s}, u(x,y) is the
bulk flow velocity {m/s} in a fracture centered at x, and d
is the spacing between fractures {m}. Time is expressed in
seconds. Equation (1) describes the horizontal transport of
gas in the bulk matrix medium at a given height y {m},
and equation (2) describes transport along a fracture where
x is taken to be 0. These equations take into account diffusion and advection as well as the radioactive decay of the
isotopes. The cross-sectional area per unit length along the
fracture is assumed for simplicity to be constant, but it could
also be included as a function of depth.
[8] A similar set of coupled differential equations is used
to determine the response of the pressure at each point in
the model, p(x,y;t) {Pa} due to a change in the surface
1
Auxiliary materials are available in the HTML. doi:10.1029/
2012GL053885.
2
LOWREY ET AL.: SUBSURFACE MASS TRANSPORT AFFECTS RADIOXENON SIGNATURES
time-dependent source term that results from the decay of iodine precursors. Low pressure spikes can have a significant
effect by drawing gas to the surface and depleting the fission
xenon. The ratios of radioxenon isotopes emitted above
ground after such low-pressure periods would then be
heavily influenced by xenon coming from radioiodine decay. Figure 2 shows several instances of this (beginning
around day 38) where the simulated xenon ratios are pushed
toward the signal that would come solely from decay of
radioiodine precursors (indicated by the dashed line). In
situations such as these, monitoring of down-wind air samples (which would detect the purged xenon) and on-site sampling (which would detect xenon from recently decayed iodine) could produce very different signals, even if the
measurements were made on the same day. Importantly,
after 10 days postdetonation, none of the simulated data
coincide with the radioxenon signal from a commercial
light water reactor, which makes it easy to rule this source
out during an on-site inspection if all four radioxenon isotopes of interest are measured.
[17] Xenon ratios that fall to the left and right of the iodine
and fully fractioned lines can be explained in terms of differential transport. Decreasing atmospheric pressure will
increase the rate of xenon movement into fractures as well
as its upward convection within them. The individual xenon
isotopes can be thought of as comprising separate, overlapping plumes. The rate at which isotopes diffuse through
the geology is inversely proportional to the square root of
their mass [Bird et al., 1960]. As a result, lighter xenon isotopes will travel faster than do the heavier ones. While the
difference in diffusion rates is small, it will cause the leading
edge of the isotope plumes to reach the surface at slightly
different times. Because isotope concentrations can vary by
orders of magnitude across a plume’s leading edge, this
can significantly skew the isotope ratios, pushing the
133m
Xe/131mXe and 135Xe/133Xe signals to the left of the
iodine line. Increases in pressure would correspondingly
force gases back down the fractures. Rapid fluctuations in
atmospheric pressure can then set up a situation where lighter
isotopes are preferentially depleted from the geology, which
would push the 133mXe/131mXe and 135Xe/133Xe signals to
the right of the fully fractioned line. In both cases the effect
would be most pronounced during the first few days after a
detonation, when the location of a plume’s leading edge is
most important, which is what is seen in Figure 2.
[18] It is clear from the simulations that geological transport of xenon gas can significantly affect the isotopic ratios
that are used to determine whether or not a clandestine
nuclear test has taken place. Critically, our work shows that
the radioxenon signal from a 26 March 1992 test would have
met the previously reported criteria for a nuclear weapon
only if the test had taken place at certain locations within
the Nevada Test Site. Although much of the simulation data
fall within the expected range, there are many instances in
which radioxenon isotope ratios are well outside of the standard domain. Verification of a nuclear weapons test under
the Comprehensive Nuclear-Test-Ban Treaty [1996] can
only be done through the detection of anthropogenic isotopes. The effect that geological transport has on radioxenon
isotope ratios needs to be considered when using these data
to determine whether or not a test has taken place. Importantly, the results of the simulations presented here show that
the region between fully fractioned and nonfractioned curves
Table 1. Range of Radioxenon Transport Parameters for the
Nevada Test Site and Simulation Seta
Detonation Depth (m)
Medium Porosity
Medium
Permeability (m–2)
Fracture Spacing (m)
Fracture Width (mm)
Nevada Test Site
Parameter Range
Parameter Set
for Simulations
450–600
0.01–0.05
and 0.35–0.45
1e-17 to 1e-15
{450, 525, 600}
{0.01, 0.05, 0.1,
0.3, 0.37, 0.45}
{1e-17, 1e-16, 1e-15}
1.0–15.0
0.005–1.5
{1.0, 2.5, 5.0, 10, 15}
{0.01, 0.1, 0.5, 1.0, 2.0}
a
Two ranges are given for the porosity of the medium and correspond to
regions of clay and granite within the Nevada Test Site.
used to compute the integral term in equation (2), which is
solved to yield the isotopic concentrations in the fracture.
The bulk flow velocities in the system are computed using
the matrix pressures, which are solved for in both the matrix
and the fractures using equations (3) and (4). Additional
details on the computational implementation of equations
(1)–(4), mesh reduction study, and benchmarking can be
found in the auxiliary material.
[13] Implementation of equations (1)–(4) requires information on the depth of detonation, initial isotope concentration, fracture width, and spacing as well as the porosity and
conductivity of the matrix material. A boundary pressure at
the surface is also required (a no-flux boundary condition
is assumed at the bottom of the simulated geology). Detailed
knowledge of the geology at the location of a suspected test
site may be difficult to obtain. However, considerable information on the range of porosity, conductivity, fracture
width, and fracture spacing are available for the Nevada Test
Site (Table 1) [McCord, 2007]. Hourly atmospheric pressure
data for the Test Site can be compiled from the weather history at the Desert Rock airfield in Mercury, Nevada. Interpolation is used to provide resolution at 1 min intervals.
[14] The radioisotopes at time t = 0 (immediately after detonation) were assumed to be contained in a region of contaminated matrix. The initial concentration of each tracked
isotope was determined by dividing its total quantity in Ci
by the approximate volume of material vaporized in a 1 kt
nuclear explosion [Carman et al., 2002]. The contaminated
matrix in the model is the region between the bottom of
the system (450 m depth) up to the depth of the fresh air
buffer (a variable) and the initial isotope concentrations
within this region were assumed to be uniform.
3. Results and Discussion
[15] We performed 990 separate simulations, each for a
different combination of transport parameters within the
Nevada Test Site. The barometric data used in the simulations were chosen to coincide with the 55 days succeeding
the 26 March 1992 US test of a fission device at Pahute
Mesa. Isotopic ratios were compiled from successive simulated 24 h outflow averages of xenon gas that reached the
surface. Figure 2 shows the results, color-coded by the number of days postdetonation, along with the atmospheric pressures that were recorded on each day. The wide range in isotopic ratios can be explained by source mixing and the
subsurface differential transport of xenon gas.
[16] The isotopic signal seen above ground is the result of
xenon that was produced directly by the fission event and a
3
LOWREY ET AL.: SUBSURFACE MASS TRANSPORT AFFECTS RADIOXENON SIGNATURES
Figure 2. Subsurface transport after an underground nuclear weapons test can cause radioxenon signatures to fall outside
of current detection criteria. Predicted xenon ratios generated from a belowground nuclear test, taking into account subsurface transport phenomena. Each data point corresponds to xenon signatures that would result from detonations within different geological conditions present within the Nevada Test Site, given the atmospheric pressures that were recorded
during the 55 days that followed the 26 March 1992 weapons test at Pahute Mesa within the Nevada Test site. Approximately 20% of the simulated conditions produced signatures that fall outside of the current boundaries for detection of such
a test. Large low pressure spikes, or extended periods of low pressure, push the simulated xenon ratios toward the signal
from decay of radioiodine precursors (shown by the dashed line). It is clear that the current assumption of what constitutes
a radioxenon signal for a belowground nuclear weapons test is too narrowly defined. The full region between the radioiodine
and fully fractionated signals should be considered when evaluating data on suspected nuclear tests.
Bowyer, T. W., et al. (2002), Detection and analysis of xenon isotopes for
the comprehensive nuclear-test-ban treaty international monitoring system, J. Environ. Radioact., 59(2), 139–151.
Carman, A. J., J. I. McIntyre, T. W. Bower, J. C. Hayes, T. R. Heimbinger,
and M. E. Panisko (2002), Discrimination between anthropogenic sources
of atmospheric radioxenon, Trans. Am. Nucl. Soc., 87, 89–90.
Carrigan, C., R. Heinle, G. Hudson, J. Nitao, and J. Zucca (1996), Tracer
gas emissions on geological faults as indicators of underground nuclear
testing, Nature, 382, 528–531.
Carrigan, C., R. Heinle, G. Hudson, J. Nitao, and J. Zucca (1997), Barometric gas transport along faults and its application to nuclear test-ban monitoring Rep. UCRL-JC-127585, US Department of Energy.
Chen, Z. X. (1989), Transient flow of slightly compressible fluids through
double-porosity, double-permeability systems, Transp. Porous Media,
4, 147–184.
Gringarten, A. C. (1984), Interpretation of tests in fissured and multilayered
reservoirs with double-porosity behavior, J. Pet. Technol., 36, 549–564.
Hannon, W. J. (1985), Seismic Verification of a Comprehensive Test Ban,
Science, 227, 251–257.
Kalinowski, M., et al. (2010), Discrimination of Nuclear Explosions against
Civilian Sources Based on Atmospheric Xenon Isotopic Activity Ratios,
Pure Appl. Geophys., 167(4), 517–539.
Le Petit, G., P. Armand, G. Brachet, T. Taffary, J. P. Fontaine, P. Achim, X.
Blanchard, J. C. Piwowarczyk, and F. Pointurier (2008), Contribution to
the development of atmospheric radioxenon monitoring, Journal of Radio and Nuclear Chemistry, 276(2), 391–398.
Lowrey, J. D., S. R. Biegalski, and M. R. Deinert (2012), UTEX modeling
of radioxenon isotopic fraction from subsurface transport, J. Radioanal.
Nucl. Chem., doi:10.1007/s10967-012-2026-1.
McCord, J. (2007), Phase I Contaminant Transport Parameters for the
Groundwater Flow and Contaminant Transport Model of Corrective
Action Unit 97: Yucca Flat/Climax Mine, Nevada Test Site, Nye
County, Nevada, Revision 0 Rep. S-N/99205--096.
Neretnieks, I., and A. Rasmuson (1984), An approach to modeling radionuclide migration in a medium with strongly varying velocity and block
sizes along the flow pat, Water Resour. Res., 20(12), 1823–1836.
Nilson, R. H., and K. H. Lie (1990), Double-porosity modeling of oscillatory gas motion and contaminant transport in a fractured porous medium,
Int. J. Numer. Anal. Methods Geomech., 14, 565–585.
Nilson, R. H., P. E. Peterson, K. H. Lie, N. R. Burkhard, and J. R. Hearst
(1991), Atmospheric pumping: a mechanism causing vertical transport
is too narrowly defined to encompass the xenon signals that
could result from a nuclear test. The xenon signature of a
belowground test is more appropriately bounded by the fully
fractioned curve and the signal that would come solely from the
decay of iodine precursors. Radioxenon ratios that sit between
the radioiodine and nonfractioned signal curves should be considered when evaluating data on a suspected nuclear test.
[19] Acknowledgments. This material is based upon work supported
by the Department of Energy, National Nuclear Security Administration,
under Award Number DE-AC52-09NA28608. Special thanks to Geoff
Recktenwald for discussions about simulation methods, to Charles Carrigan
for discussions about subsurface transport of noble gases and to William H.
Press and Sara L. Sawyer for editorial comments and suggestions.
Disclaimer: “This report was prepared as an account of work sponsored
by an agency of the United States Government. Neither the United States
Government nor any agency thereof, nor any of their employees, makes
any warranty, expressed or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information,
apparatus, product, or process disclosed, or represents that its use would
infringe privately owned rights. Reference herein to any specific commercial
product, process, or service by name, trademark, manufacturer, or otherwise
does not necessarily constitute or imply its endorsement, recommendation
or favoring by the United States Government or any agency thereof.”
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5
Supplemental information.
∂C( x, y ) i ∂
∂C ( x, y ) i ⎞
∂⎛
⎟ − φ λ C ( x, y ) i φm
+ C ( x, y ) i v ( x, y ) = ⎜φ m Di
∂x
∂x ⎝
∂t
∂x ⎠ m i
(
φf
∂C(x, y )i ∂
+ C (x, y )i u(x, y ) =
∂y
∂t
δ
∫
)
2
0
(
"!
)
∂C(x, y )i
∂C(x, y )i ⎞
∂⎛
⎟ − φ λ C(x, y )i
φm
dx + ⎜ φ f Di
∂y ⎝
∂t
∂y ⎠ f i
#!
Here φm and φf are the matrix and fracture porosities {dimensionless}, C(x,y)I is the
concentration of the i’th isotope {Ci/m3}, v(x,y) is the bulk flow velocity {m/s} through the
matrix, Di {m2/s} is the diffusion coefficient of the i’th xenon isotope, λi is its decay
constant {1/s}, u(x,y) is the bulk flow velocity {m/s} in a fracture centered at x, and δ is
the spacing between fractures {m}. Time is expressed in seconds. Equation (1)
describes the horizontal transport of gas in the bulk matrix medium at a given height y
{m}, and Eq. (2) describes transport along a fracture where x is taken to be 0, Fig. S1.
"#!
"#!
"
These equations take into account diffusion and advection as well as the radioactive
decay of the isotopes. The cross-sectional area per unit length along the fracture is
assumed for simplicity to be constant, but it could also be included as a function of
depth.
The diffusion coefficients Di appearing in Eqs. (1) and (2) are tortuosity-weighted
according to:
Di =
φ
D
τ i ,0
(3)
where the tortuosity τ {dimensionless} and porosity ϕ have separate values within the
fracture and matrix media (1-3).
The flow velocities, u(x,y) and v(x,y) for bulk motion along the vertical fracture and within
the matrix are functions of the differential pressures within the cracks that arise from
variations in atmospheric pressure. A similar set of coupled differential Eqs. (4) and (5)
is used to determine the response of the pressure at each point in the model, p(x,y;t)
{Pa} due to a change in the surface pressure at time t. Here we use a semicolon to
indicate that the symbol to its right is held constant:
∂p(x;y ) ∂ ⎛ ∂p(x;y )⎞
= ⎜α m
⎟
∂t
∂x ⎠
∂x ⎝
∂p(0, y ) φ m
=
∂t
φf
∫
δ
2
0
∂p( x, y )
∂ ⎛ ∂p(0, y ) ⎞
dx + ⎜ α f
⎟
∂y ⎝
∂t
∂y ⎠
(4)
(5)
The notation p(0,y) indicates that Eq. (5) represents the rate of change of pressure in the
fracture, where the coordinate x is equal to zero. The pneumatic diffusivities of the
matrix medium and fracture, αm and αf, {m2/s} are assumed to be constant as a function
of depth and are defined by:
δ 2f p0
αf =
12 μφ f
and
αm =
km p0
μφ m
(6)
where p0 is the mean pressure of the system {Pa}, μ is the dynamic viscosity of air
{Pa⋅s}, and km is the permeability of the bulk matrix medium {m2}. Then the fluid
velocities that result are given by:
k ∂p( x, y )
v ( x, y ) = m
μ ∂x
and
δ 2f ∂p( x, y )
u( y ) =
12 μ ∂y
(7)
Equations (1-7) are discretized by using first-order backwards differencing for first-order
derivatives and second-order centered differencing for the second-order diffusive
operators. The discretized equations are formulated as a set of tridiagonal matrix
equations. Solution of these equations for each time step is made by Gaussian
elimination with periodic boundary conditions at the interior of the matrix and a closed
bottom boundary. It is assumed that there is no interaction between vertical layers in the
matrix, thus Eqs. (1,2,4,5) may be solved separately for each layer, and for the fracture.
Equation (1) is solved at every layer to find the isotopic concentrations within the matrix.
The concentrations are used to compute the integral term in equation (2). Equation (2)
is solved to yield the isotopic concentrations in the fracture. The bulk flow velocities in
the system are computed using the matrix pressures, which are solved for in the matrix
and fracture using Eqs. (4) and (5).
The tridiagonal matrix method used to solve Eqs. (1,2,4,5) was verified in three separate
ways: i) a comparison with analytically obtained results, ii) testing for convergence in a
mesh reduction study and iii) a comparison with alternate numerical methods.
The radioisotopes immediately after detonation were assumed to be contained in a
region of contaminated matrix, Fig. S1. The initial concentration of each tracked isotope
was determined by dividing its total quantity in Ci by the approximate volume of material
vaporized in a 1kt nuclear explosion [5]. The contaminated matrix in our model is the
region between the bottom of the system (450m depth) up to the depth of the fresh air
buffer (a variable) and the initial isotope concentrations within this region were assumed
to be uniform.
A mesh reduction study was conducted and it was found that a time step of 60 seconds
along with horizontal and vertical grid spacings of 0.05m and 4.5m were sufficient for
convergence of isotope concentrations to the 5th decimal place. The surface pressure
variations used were sampled at one hour intervals at Mercury, Nevada and linear
interpolation was used to produce data with 60 second resolution.
Comparison With Alternate Numerical Methods. An independent solution for Eqs. (1-2,
4-5) was computed using Newton's method. Equations (4) and (5) are coupled via the
interaction between the fracture and the matrix. The pressures converged after fewer
than six iterations, and Eqs. (4) and (5) were satisfied to machine precision. A
simultaneous solution method was also implemented Eqs. (1-4) which formulated the
fracture-matrix system as a large, sparse block-diagonal matrix. The M x N pressure
matrix was written as a column vector p of length MxN, and the matrix equation Ap = f
was solved using Gaussian elimination.
In all cases, the computed pressures agreed to within machine precision. Similarly, Eqs.
(1,2) were discretized and formulated as a block-diagonal matrix. The concentrations
computed using this method and using the tridiagonal Gaussian elimination method
agreed to a precision attributable to round off error and approaching machine precision
as well.
Supplemental References
1.
2.
3.
4.
5.
Z. X. Chen, Transport in Porous Media 4, 147 (1989).
A. C. Gringarten, Journal of Petroleum Technology 36, 549 (1984).
I. Neretnieks, A. Rasmuson, Water Resources Research 20, 1823 (1984).
R. H. Nilson, E. W. Peterson, K. H. Lie, Journal of Geophysical Research 96,
21933 (1991).
P.J. Closmann, Journal of Geophysical Research 74, 15 (1969).