TY - JOUR
T1 - A direct simulation demonstrating the role of spacial heterogeneity in determining anomalous diffusive transport
AU - Voller, Vaughan R.
N1 - Publisher Copyright:
© 2015. American Geophysical Union. All Rights Reserved.
Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.
PY - 2015/4/1
Y1 - 2015/4/1
N2 - Typically the spreading length-scale in a diffusion process increases in time as ℓ∼tn,n=1/2. In the presence of spatial heterogeneities, however, so-called anomalous diffusion can occur, here the time exponent n≠1/2. The objective of this paper is to present a numerical simulation that directly demonstrates the link between spacial heterogeneity and anomalous diffusion. The simulation is of the infiltration of a fluid into a horizontal unit area Hele-Shaw cell in which the permeability at specified locations, laid out as a Sierpinski fractal carpet pattern, can differ from the nominal value K = 1. When there is no permeability contrast, the fluid infiltration has a diffusion-like behavior, i.e., the filled plan-form area increases in time as F=Atn,n=1/2. When there is a permeability contrast, however, although the evolution of infiltration still follows a power law, anomalous behavior is observed; subdiffusive (n<1/2) when K < 1 and superdiffusive (n>1/2) when K > 1. These anomalous behaviors persist, even when the permeability contrast is only imposed over the largest sized fractal pattern element. But, if the pattern over which the permeability contrast is imposed has a subdomain length scale (obtained by filling in the largest sized element in the Sierpinski carpet with the smaller sized elements), normal diffusion n=1/2 behavior is recovered. In the special case that the imposed permeability in the pattern elements is K蠐1, an approximate model, directly relating the coefficient and exponent in the infiltration power law to the porosity and fractal dimension of the carpet pattern, is derived and validated.
AB - Typically the spreading length-scale in a diffusion process increases in time as ℓ∼tn,n=1/2. In the presence of spatial heterogeneities, however, so-called anomalous diffusion can occur, here the time exponent n≠1/2. The objective of this paper is to present a numerical simulation that directly demonstrates the link between spacial heterogeneity and anomalous diffusion. The simulation is of the infiltration of a fluid into a horizontal unit area Hele-Shaw cell in which the permeability at specified locations, laid out as a Sierpinski fractal carpet pattern, can differ from the nominal value K = 1. When there is no permeability contrast, the fluid infiltration has a diffusion-like behavior, i.e., the filled plan-form area increases in time as F=Atn,n=1/2. When there is a permeability contrast, however, although the evolution of infiltration still follows a power law, anomalous behavior is observed; subdiffusive (n<1/2) when K < 1 and superdiffusive (n>1/2) when K > 1. These anomalous behaviors persist, even when the permeability contrast is only imposed over the largest sized fractal pattern element. But, if the pattern over which the permeability contrast is imposed has a subdomain length scale (obtained by filling in the largest sized element in the Sierpinski carpet with the smaller sized elements), normal diffusion n=1/2 behavior is recovered. In the special case that the imposed permeability in the pattern elements is K蠐1, an approximate model, directly relating the coefficient and exponent in the infiltration power law to the porosity and fractal dimension of the carpet pattern, is derived and validated.
KW - anomalous diffusion
KW - direct simulation
KW - fractal carpet
KW - spacial heterogeneities
UR - http://www.scopus.com/inward/record.url?scp=85027923349&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85027923349&partnerID=8YFLogxK
U2 - 10.1002/2014WR016082
DO - 10.1002/2014WR016082
M3 - Article
AN - SCOPUS:85027923349
VL - 51
SP - 2119
EP - 2127
JO - Water Resources Research
JF - Water Resources Research
SN - 0043-1397
IS - 4
ER -