Not NORMAL: Anomalous transport in hydrology, hydrogeology, and geomorphology
Rina SchumerDivision of Hydrologic Sciences
Desert Research InstituteReno, NV, USA
CUAHSI Webinar - March 24, 2017
Collaborators:
Antoine Aubeneau, David Benson, Boris Baeumer, Diogo Bolster, Tyler Doane,
Jen Drummond, Noah Finnegan, Efi Foufoula-Georgiou, David Furbish, Hal
Voepel, Marwan Hassan, Doug Jerolmack, Raleigh Martin, Brandon McElroy,
Mark Meerschaert, Aaron Packman, Gary Parker, Anna Pelosi, Matt Reeves,
Josh Roering, Alessandro Taloni, Yong Zhang
Summer Institutes Maki Chair in Hydrologic
Sciences
With apologies to the long list of relevant references not shown on slides!
• Sediment transport
• fluvial
• hillslope
• Solute transport
• porous media
• fracture networks
• streams
• Surface evolution
• streambed elevation
• elevation of a fluctuating surface
• hillslope evolution
• Dispersal of seeds
• Wildfire propagation
• Bacterial motion in porous media
• Water
• unsaturated flow
• fracture flow
Transport in the environment
2
2
P PD
x
P
xtv
We use equations that take this form
….a lot.
Instead of P, we might see
h=hydraulic head
C=concentration
T=temperature
h=elevation
g=activity
The way I learned the diffusion equation
CF D
x
FC
t x
2
2
C CD
t x
Constitutive
equation for fluxConservation of mass
Diffusion equation
What I didn’t learn about the diffusion equation
CF D
x
FC
t x
2
2
C CD
t x
Constitutive
equation for flux Conservation of mass
Diffusion equation
Lots of
assumptions about
particle motion go
into this equation
There are
assumptions here
that are easily
violated
x x
z
y
x
Overview
• Normal transport vs anomalous transport
• Anomalous transport in hydrologic and geomorphic applications
• Implications of anomalous transport for hydrology and geosciences
Anomalous transport = Not normal transport
What is normal?
code modified from Brett Sanders, UCI
“particles” represent any quantity that is actively or passively transported:
solute, sediment, earth surface, bacteria, water, molecules, seeds, heat, etc.
microscopic
motion
Macroscopic
(bulk, time averaged)
observables
tx x
Normal, diffusive transport
Particle spread
(as determined by the variance
or msd of particle location)
grows linearly with time
2( ) 2x t Dt
The spread of particles with time is
a Gaussian or “normal” probability
density function
t
2D
Normal, diffusive transport
Particle spread
(as determined by the variance
or msd of particle location)
grows linearly with time
2( ) 2x t Dt
solution describing the spread of
particles with time is a Gaussian or
“normal” probability density
function
These characteristics define
NORMAL transport
Everything else is
ANOMALOUS2D
x t
Anomalous transport
Particle spread
(as determined by the variance
or msd of particle location)
grows non-linearly with time
2( )x t Dt
solution describing the spread of
particles with time is non-Gaussian
(not normal) probability density
function
Anomalous diffusion
typically refers to a
power-law form
subdiffusion
diffusion
superdiffusion
1
1 1
Derivation of transport equations
Deterministic
particle motion
F=ma
Unmeasurable
Motion can be
broken into a
series of jumps.
Jump lengths
(and time) can
be described
statistically
take a limit
Statistical
conceptual
model
( ) ( )2
2
, ,D
P x t P x t
t x
¶ ¶=
¶ ¶
A gain-loss eqn for
the probability a
particle will at
each location
Discrete stochastic model
Fokker Planck
equations,
e.g.
Particle motion required to get normal transport
Conditions for normal transport:
1) Independence of individual particles
2) Statistically independent displacements
(at some characteristic timescale)
3) At this timescale, displacements have a
well-defined mean and variance
LOTS of anomalous transport models describing particle
motion that breaks one or more of these rules
Violations of normal transport conditions
How to violate normal transport conditions:
1) Independence of individual particles
2) Statistically independent displacements
(at some timescale)
3) At this timescale, displacements have a well-
defined mean and variance
LOTS of anomalous transport models describing particle
motion that breaks one or more of these rules
Power-law distributed
immobile periodsPower-law distributed velocity
distribution
correlated immobile periods
correlated velocity and immobile
periods
correlated velocity
Time dependent dispersion Space-dependent dispersion
Single file
trajectories,
crowding
(mostly
biophysics)
LOTS of anomalous transport models describing particle motion that
breaks one or more of the normal rules
Table 1 from Metzler et al, 2014 PCCP
Normal diffusion
Historical note – original application
2( ) 2x t Dt
#
gas constT
Boltzmann Const T AvagadroD
mass friction coeff mass friction coeff
Smoluchowski Einstein
pollen grain being kicked
around by vibrating water
molecules
t
In the original application, D arises from
“random” thermal fluctuations
2D
Environmental applications
In hydrologic transport, velocity fluctuations
arise from environmental heterogeneity.
Holy grail in each subfield is to identify
1. What is the scaling parameter, ?
2. What are the physical controls on ?
3. What is the dispersion coefficient, D?
4. What are the physical controls on D?
1
Log t
Log
2( )x t Dt
on a log-log plot, is the slope, D is a shift :2log ( ) log logx t D t
Signs of anomalous transport(and how anomalous transport is related to fractals)
“Scale dependence of dispersivity”: D is not constant, but a function of t or x.
Image from http://fractalfoundation.org/OFC/OFC-10-4.html
Length of the coastline of Britain varies with measurement interval
because we measure a fractal with a linear ruler.
Anomalous transport analog:
Particles moving in heterogeneous environments take fractal paths:
Estimate D using a linear ruler 2 ( )
2
x tD
t
Classic fractal example:
2( )x t Dt
Use of anomalous transport models
Statistics of
individual
particle
motions
Long time
average of
bulk particle
motion
independent vs correlated
jump length distribution
waiting time distribution
etc.
scaling rate
diffusion/dispersion coefficient
2( )x t Dt
What can we measure? (individual particle motion or bulk behavior?)
Can we observe the transition across a characteristic scale?
2 Dtt
• Sediment transport
• fluvial
• hillslope
• Solute transport
• porous media
• fracture networks
• streams
• Surface evolution
• streambed elevation
• elevation of a fluctuating surface
• hillslope evolution
• Dispersal of seeds
• Wildfire propagation
• Bacterial motion in porous media
• Water
• unsaturated flow
• fracture flow
Anomalous transport in the environment
Immobile (bCim)
Total (Ctot)
Mobile (Cm)
0 5 10 15 20 25 30 35 400
0.05
0.1
0.15
0.2
Distance from source (x)
Rela
tive
mas
s in
eac
h p
has
e
Solute transport in porous media
What can we measure?
Breakthrough curves and spatial snapshots of tracer studies (slope of
late-time breakthrough curve is related to .
Signs of anomalous transport
Skewed breakthrough curves with long-late time tails; incomplete mass
recovery
Anomalous transport attributed to:
Power-law residence time of solute in immobile zones; correlation in flow
field
Drivers:
lnK-distribution around preferential flow paths
Correlation in hydraulic conductivity field
Flow in fractures
What can we measure?
Some lab (mostly numerical)
Signs of anomalous transport
Time dependent dispersion coefficient
Anomalous transport attributed to:
Small-scale eddies within fractures due to surface heterogeneity
Drivers:
proportional to variance of fracture aperture (Wang and Cardenas, 2014 WRR)
flow direction
Coarse sediment transport*
What can we measure?
Individual particle motion in flumes or tagged in the field;
Particle residence time by continuously measuring bed elevation
Signs of anomalous transport
Observation of long particle residence time as sediment vertically mixes into bed;
Power-law decreasing virtual velocity
Anomalous transport attributed to (Bialik et al. 2015 JFM):
1. Initial acceleration of entrained particles
2. Correlated motion of continuously transported sediment
3. Incorporation of power-law distributed rest periods
*Anomalous transport equations based on
entrainment rather than flux-based perspective(Lots of Furbish and Parker papers; also see work by Ancey
and Nikora groups)
Hans Albert Einstein
Bedload transport as a probability problem
Use caution when interpreting tracked particles(sediment, bacteria, numerical)
2 2( )x t t
Mean squared
displacement
(variance) of a
plume of particles
Time averaged
mean squared
displacement of a
single particle
?
Ergodic hypothesis
This is valid for the normal diffusion model,
but not all anomalous transport models
Be careful when tracking particles…
not all anomalous diffusion models are ergodic
Modified Table 1 from Metzler et al, 2014 PCCP
Normal diffusion
No
No
No
No
Yes
Yes
Yes
Yes
No
No
No
No
No
No
No
No
No
No
No
Ergodic?
MSD for bulk
(ensemble motion)
MSD for tracked
particles
Anomalous solute transport in fluvial systems
Gomez et al., 2012 (WRR)
http://susa.ston
edahl.com/res
earch.html
What can we measure?
Breakthrough curves of tracer studies (long time first passage)
Signs of anomalous transport
Skewed breakthrough curves; incomplete mass recovery
Anomalous transport attributed to:
Power-law residence time of solute in hyporheic zone
(immobile zone compared with in-channel flow)
Drivers:
Bedform induced exchange;
various length subsurface flowpaths
Co
nce
ntr
atio
n
Haggerty et al., 2000 (WRR)
Anomalous suspended sediment transport(catchment scale)
http://nwrm.eu/measure/floodplain-
restoration-and-management
What can we measure?
Turbidity (hard to do tracer studies at large scales)
Signs of anomalous transport
Filtered downstream signal of fine sediment pulses
Anomalous transport attributed to:
Power-law residence time of fine sediment in flood plains
Drivers:
Recurrence of sediment remobilization events
Return times to the surface of buried sediment
Hillslope evolution
Foufoula-Georgiou et al., 2010 JGR
What can we measure?
Hillslope profiles and their time evolution
Signs of anomalous transport
Poor fit of profiles to local or non-linear alternatives
Anomalous transport attributed to:
Non-local transport (entrainment and travel distance are not
simply functions of local slope)
Drivers:
Vegetation and barriers that act as sediment capacitors
Various transport processes that act over different scales
Doane et al., in prep
Generation of the geologic record(transport of the earth surface up and down by erosion and deposition)
What can we measure?
Bed thickness, linear rates of accumulation
Signs of anomalous transport
Sadler Effect: rates of accumulation calculated from
cores is a function of measurement interval t
“Anomalous” geologic record attributed to:
Incomplete geologic record; power-law stratigraphic
hiatuses
Drivers:
Long-range correlation and anomalous sediment
transport across a fluctuating earth surface lead to
power-law missing periods of time in cores
Implications of anomalous transport
1. Increased focus on residence time or travel time distributions (TTD)
2. Rethinking of reactive transport (reactions and anomalous transport strongly
coupled)
3. When do we reach characteristic/length and time scales and how do we
interpret them?
4. Advances in random walk/ Monte-Carlo simulations and numerical methods
for non-local transport
Why hasn’t anomalous transport gone mainstream?
1. Idea that if we discretize finely enough, heterogeneity can be captured
2. Extra parameters
3. Still in the discovery phase by application
4. Non-stationarity
5. Lack of textbook-level material
Summary
• In hydrology and earth surface applications, normal transport is not normal
• Particles in heterogeneous environments take fractal paths whose properties appear scale dependent when measured with a linear ruler
• Solute and fine particle transport are observed as bulk properties usually related to velocity correlations and power-law residence time distributions
• Coarse sediment, bacteria, seeds are amenable to single particle trajectory analysis (where care must be taken)
• State of the art is identification of scaling rates () and D
• Big implications for reaction and processing
