IDAEA-CSIC, Barcelona, SpainUPSCALED CONTINUOUS RANDOM TIME WALK MODEL I We consider...
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TRANSPORT UNDER ADVECTIVE TRAPPING
Juan J. Hidalgo1, Insa Neuweiler2, and Marco Dentz1
1IDAEA-CSIC, Barcelona, Spain
2Leibniz Universität Hannover, Hannover, Germany
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INTRODUCTION
Motivation:I Advective trapping occurs when solute enters low velocity zones in
heterogeneous porous media.I Classical approaches combine slow advection and diffusion into a dispersion
coefficient or a single memory function.ObjectiveI We investigate advective trapping in homogeneous media with low
permeability circular inclusions.I We build an upscaled model in the continuous time random walk framework.
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FLOW PROPERTIES
Mean velocity in the matrix isproportional to the area occupied bythe inclusions χ.Velocity in the inclusions is notconstant. The mean velocity in theinclusions vi is log-normallydistributed and proportional to χ.
FIGURE 1: Streamlines in a mediumwith randomly placed inclusions.
0
150
300
0.01 0.02 0.03 0.04 0.05 0.06
p(v)
v
vi
vi
10−1
100
101
102
103
0.01 0.02 0.03 0.04 0.05 0.06 0.07
p(vi)
vi
r0r0/2r0/3r0/4
0
0.1
0.2
0.3
0.4
0.5
χ
FIGURE 2: Velocity distribution inside theinclusions (top) and mean velocitydistribution (bottom).c© AUTHORS. ALL RIGHTS RESERVED
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TRANSPORT PROPERTIES
The breakthrough curves reflect thetrapping of particles in the lowpermeability inclusions.The trapping rate follows a Poissondistribution.
FIGURE 3: Transport through a mediumwith randomly placed inclusions.
10−5
10−4
10−3
10−2
10−1
100
0 10 20 30 40 50
f(t)
t
xc
ℓc5ℓc10ℓc
(a)
10−5
10−4
10−3
10−2
10−1
200 250 300 350 400 450 500 550 600
f(t)
t
xc
300ℓc387ℓc
(c)
10−4
10−3
10−2
10−1
100
0 5 10 15 20
p(ntr)
ntr
xc
10ℓc50ℓc200ℓc387ℓc
FIGURE 4: Breakthrough curves at increasingdistance from the inlet and advection -dispersion equation solution fit (top). Thenumber of trapping events follows a Poissondistribution (bottom).c© AUTHORS. ALL RIGHTS RESERVED
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UPSCALED CONTINUOUS RANDOM TIME WALK MODEL I
We consider advective-dispersive particle transitions in the mobile matrix
dx(s) = vmatrixds +√
2Dmdsξ(s),
with s the mobile time spend outside the inclusions, Dm is diffusion (fromEames & Bush, 1999) and ξ(s) a Gaussian white noise.During the mobile time s particles encounter ns inclusions. The clock time t(s)after the mobile time s has passed is given by
t(s) = s +ns
∑i=1
τi
where ns is Poisson distributed and the trapping times τi depend on thedistance (random uniform) and the velocity at the visited inclusion (randomlog-normal)
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UPSCALED TRANSPORT MODEL RESULTS
10−5
10−4
10−3
10−2
10−1
100
0 10 20 30 40 50
f(t)
t
xc
ℓc5ℓc10ℓc
(a)
10−5
10−4
10−3
10−2
10−1
200 250 300 350 400 450 500 550 600
f(t)
t
xc
300ℓc387ℓc
(c)
FIGURE 5: Breakthrough curves at increasing distance from the inlet (dots) and upscaledCTRW model results 8solid line).
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SUMMARY & CONCLUSIONS
Purely advective transport was here considered as a limiting case foradvective-diffusive transport.The shape of breakthrough the curves cannot be predicted with amacrodispersion coefficient.We developed a CTRW model developed parameterized by measurablemedium properties: the trapping rate (Poisson distributed), the velocity inthe matrix (a function of χ) and the mean velocity distribution inside theinclusions (log-normal).
Acknowledgements
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