Spatial Markov processes for modeling Lagrangian particle dynamics in heterogeneous porous media.

Abstract

We investigate the representation of Lagrangian velocities in heterogeneous porous media as Markov processes. We use numerical simulations to show that classical descriptions of particle velocities using Markov processes in time fail because low velocities are much more strongly correlated in time than high velocities. We demonstrate that Lagrangian velocities describe a Markov process at fixed distances along the particle trajectories (i.e., a spatial Markov process). This remarkable property has significant implications for modeling effective transport in heterogeneous velocity fields: (i) the spatial Lagrangian velocity transition densities are sufficient to fully characterize these complex velocity field organizations, (ii) classical effective transport descriptions that rely on Markov processes in time for the particle velocities are not suited for describing transport in heterogeneous porous media, and (iii) an alternative effective transport description derives from the Markovian nature of the spatial velocity transitions. It expresses particle movements as a random walk in space time characterized by a correlated random temporal increment and thus generalizes the continuous time random walk model to transport in correlated velocity fields.

Full text

Page 1

Spatial Markov processes for modeling Lagrangian particle dynamics
in heterogeneous porous media
Tanguy Le Borgne*
Géosciences Rennes, UMR 6118, CNRS, Université de Rennes 1, Rennes, France
Marco Dentz
Department of Geotechnical Engineering and Geosciences, Technical University of Catalonia (UPC), Barcelona, Spain
Jesus Carrera
Institute of Environmental Analysis and Water Studies (IDAEA), CSIC, Barcelona, Spain
?Received 28 April 2008; published 26 August 2008?
We investigate the representation of Lagrangian velocities in heterogeneous porous media as Markov pro-
cesses. We use numerical simulations to show that classical descriptions of particle velocities using Markov
processes in time fail because low velocities are much more strongly correlated in time than high velocities. We
demonstrate that Lagrangian velocities describe a Markov process at fixed distances along the particle trajec-
tories ?i.e., a spatial Markov process?. This remarkable property has significant implications for modeling
effective transport in heterogeneous velocity fields: ?i? the spatial Lagrangian velocity transition densities are
sufficient to fully characterize these complex velocity field organizations, ?ii? classical effective transport
descriptions that rely on Markov processes in time for the particle velocities are not suited for describing
transport in heterogeneous porous media, and ?iii? an alternative effective transport description derives from the
Markovian nature of the spatial velocity transitions. It expresses particle movements as a random walk in space
time characterized by a correlated random temporal increment and thus generalizes the continuous time ran-
dom walk model to transport in correlated velocity fields.
DOI: 10.1103/PhysRevE.78.026308PACS number?s?: 47.56.?r, 05.40.Fb, 05.10.Gg, 05.60.?k
I. INTRODUCTION
The transport of chemicals in groundwater, the ocean, or
the atmosphere is controlled by the multiscale organization
of natural flows. Natural flow fields generally have a com-
plex organization characterized by heterogeneous structures
at different scales ?1–4?. Transport in such velocity fields
displays non-Fickian characteristics such as strong tailing of
first arrival times and non-Gaussian spatial distributions
?e.g., ?5??. Accounting for these properties is particularly
critical for the prediction of contaminant transport in the sub-
surface, which is the focus of the present study. A number of
approaches such as continuous time random walk ?CTRW?
theory and fractional Fokker-Planck equations describe
anomalous transport ?6–11?. For transport in shear flows the
role of the flow organization for effective transport can be
quantified exactly ?12?. For many natural flow scenarios
cases, however, and in particular for transport in groundwa-
ter flows, the role of the local flow organization on effective
transport is not understood quantitatively ?13,14? which lim-
its our predictive capabilities.
Let us consider the one-dimensional space-time random
walk model,
x?n+1?= x?n?+ ??n?,
t?n+1?= t?n?+ ??n?,
?1?
where both the spatial and temporal increments ??n?and ??n?
are random. Equation ?1? describes a continuous time ran-
dom walk ?CTRW? ?e.g., ?15??. The particle velocity at step
n is given by the kinematic relationship v?n?=??n?/??n?. Note
that it may include advection and diffusion processes. The
space-time positions of a particle after n steps are denoted by
?x?n?,t?n??. Non-Fickian transport can be modeled in this
framework by choosing the spatio-temporal increments from
distributions whose widths are of the order of or larger than
the observation scales. Fickian behavior is obtained if both
the widths of the distributions of the spatial and temporal
increments are small compared to the observation lengths
and times.
The application of effective random walk models to the
description of effective transport in heterogeneous velocity
fields is often based on the postulation of a joint distribution
of the spatial and temporal increments and the subsequent fit
of the model to some dispersion measurements ?such as the
evolution of the spatial moments of the solute cloud or
breakthrough curves?. However, the relationship between the
flow field organization and the distribution of spatial and
temporal increments is unknown in general. In particular, in
many effective random walk descriptions velocity correlation
properties, which carry information on the velocity field or-
ganization, cannot be included easily ?16?.
Here, we derive a joint distribution of spatial and tempo-
ral increments from the Lagrangian velocity statistics and
correlation properties instead of postulating it. The Lagrang-
ian statistics are obtained from numerical simulations of flow
and transport in highly heterogeneous porous media. La-
grangian flow characteristics play an important role for the
qualitative and quantitative understanding of non-Fickian
transport in heterogeneous flow fields ?13?. Specifically,
strong correlations have been identified as a cause for non-
*tanguy.le-borgne@univ-rennes1.fr
PHYSICAL REVIEW E 78, 026308 ?2008?
1539-3755/2008/78?2?/026308?9?
©2008 The American Physical Society026308-1

Page 2

Fickian transport patterns ?e.g., ?7??. A different source for
non-Fickian transport is broad disorder distributions ?e.g.,
?17??. Long disorder correlations on one hand and broad het-
erogeneity distributions on the other hand are possible causes
for anomalous transport patterns with different impact on the
effective transport behavior. Our ultimate objective is to de-
fine an effective transport model that is consistent with the
velocity fields organization and that is fully characterized by
the Lagrangian velocity statistics.
In this paper we analyze the Lagrangian flow characteris-
tics with the aim to obtain a simple formulation for effective
particle dynamics as a Markovian model. To this end we set
up a numerical random walk model for transport in a hetero-
geneous Darcy flow field, project the transport process on the
direction of the mean flow, and analyze the Lagrangian ve-
locity statistics along the projected particle trajectories. The
latter defines a d=1 dimensional stochastic process that turns
out to be a Markov chain for constant spatial increments, i.e.,
the velocity at a given position along the trajectory depends
only on the velocity value at the previous positions. The
transition time between two particle positions along the tra-
jectory is given by the ?constant? transition length and ?pro-
jected? flow velocity. The latter defines a correlated continu-
ous time random walk.
II. PRELIMINARY REMARKS
Advective-diffusive solute transport is described by mass
balance, i.e., the temporal change of the solute concentration
c?x,t? is balanced by the divergence of the advective and
diffusive solute fluxes. This mass balance is expressed by the
Fokker-Planck equation
??c?x,t?
?t
+ q?x? · ?c?x,t? − D?2c?x,t? = 0,
?2?
where D is the ?constant? diffusion coefficient, and the po-
rosity ? is set equal to one. Note that for simplicity here we
consider isotropic and constant diffusion. The Eulerian ve-
locity field q?x? is given as the divergence-free solution of
the Darcy equation ?18?
q?x? = − K?x? ? h?x?,
?3?
where h?x? is the hydraulic head. The hydraulic conductivity
K?x? reflects the spatial medium heterogeneity.
The Fokker-Planck equation ?2? is exactly equivalent to
the Langevin equation ?e.g., ?19??
dx?t?
dt
= q?x?t?? + ??t?,
?4?
which describes the trajectory x?t? of a solute particle in
time. Diffusion is simulated by the Gaussian white noise
??t?, which is characterized by zero mean and the variance
??i?t??j?t???=D?ij??t−t??; the angular brackets denote the
noise average. The particle distribution c?x,t? is given in
terms of the particles trajectories as
c?x,t? = ???x − x?t???.
?5?
In the following, we focus on transport in a d=2 dimensional
heterogeneous medium. The position vector x=?x,y?T.
Equation ?4? describes transport in a single realization of
the heterogeneous medium. In most practical cases, however,
a detailed medium description is not available and also not
desirable. What one seeks is an effective transport descrip-
tion that incorporates the impact of heterogeneity on effec-
tive transport in a systematic manner. The latter can be
achieved by upscaling. The aim here is to describe the effec-
tive transport behavior by using a statistical heterogeneity
characterization. To this end, the salient ?statistical? features
of heterogeneity that determine effective transport need to be
identified and quantified. Stochastic modeling provides a
systematic way to do so.
In a stochastic approach, the spatially variable hydraulic
conductivity field K?x? represents a typical realization of a
spatial stochastic process. The latter is usually characterized
by the ?functional? distribution of the log hydraulic conduc-
tivity function f?x?=ln?K?x??, P?f?x??. We consider here sta-
tionary random fields, i.e., fields for which P?f?x+L??
=P?f?x??, with L a constant translation. In the following, the
mean value f?x? is set to zero. The two-point correlation
function of f?x? then is given by
f?x?f?x?? = Cf?x − x??.
?6?
The overline denotes the ensemble average over all realiza-
tion of the spatial random process ?f?x??. The correlation
function Cf?x? here is given by the Gaussian shaped function
2exp?−x2
Cf?x? = ?f
??,
?7?
where ? is the correlation length of the log-hydraulic con-
ductivity field and ?f
velocity field is stationary as a consequence of the stationar-
ity of f?x? and can be decomposed into its ?constant? mean
value and fluctuations about it,
2its variance, ?f
2=f?x?2. The random
q?x? = q ¯ + q??x?.
?8?
The ?Eulerian? correlation of q?x? is given by
qi??x?qj??x?? ? Cij
E?x − x??.
?9?
Without loss of generality, the one-direction of the coordi-
nate system is aligned here with the mean flow direction so
that q ¯ =e1q ¯, where e1is the unit vector in one-direction.
Note that the particle movement described by Eq. ?4? de-
pends on two stochastic processes, the spatially varying ran-
dom velocity field, and the temporal noise term. The aim is
to represent effective transport as a random walk that is com-
pletely characterized by the Lagrangian velocity statistics,
i.e., by the statistics of q?x?t??. The Lagrangian velocity cor-
relation is defined by
Cij
L?t − t?? = ?qi??x?t??qj??x?t????.
?10?
The particle velocity is given by
v?t? = q ¯e1+ q??x?t?? + ??t?.
?11?
The correlation of the velocity fluctuations in the Ito inter-
pretation then reads as
?vi??t?vj??t??? = Cij
L?t − t?? + 2D?ij??t − t??,
?12?
LE BORGNE, DENTZ, AND CARRERAPHYSICAL REVIEW E 78, 026308 ?2008?
026308-2

Page 3

Here we study effective random walk descriptions of the
transport problem ?4? as given by Eq. ?1?. Let us consider the
discretized version of Eq. ?4?:
x?n+1?= x?n?+ q?x?n???t +?2D?t??n?,
?13?
t?n+1?= t?n?+ ?t,
?14?
where ??n?is a Gaussian random variable with zero mean and
unit variance. In an effective description, we want to substi-
tute the nonlinear noise term q?x?t?? by a noise term that
does not depend on the ?effective? particle trajectory but only
on the number of steps of the effective random walk. Projec-
tion of the particle motion on the direction of mean flow
yields the equation of motion
x?n+1?= x?n?+ q1?x?n???t +?2D?t??n?,
t?n+1?= t?n?+ ?t,
?15?
To represent transport in an ensemble sense, the statistics of
q1?x?t??, which depends on the particle trajectory, need to be
explored. This can be done in two different ways. The first
way is to define an equivalent description of Eq. ?15? by
x?n+1?= x?n?+ vt
Here, ?vt
is a stochastic process that is defined by the
ensemble statistics of the Lagrangian velocity field along the
projected particle trajectory at equidistant times. However,
the process ?vt
does not need to be Markovian, and in
general is not. For transport in Gaussian random shear flow
the description ?16? turns out to be an equivalent effective
random walk model ?12? with ?vt
noise. For lognormally distributed hydraulic conductivity
fields, Fiori and Dagan ?20? established an approximate ef-
fective random walk description, whose ?correlated? noise is
characterized by the first-order approximation of the La-
grangian correlation function Cij
log-hydraulic conductivity field.
If it is possible to define a ?finite? characteristic transition
length over which the velocity field is approximately con-
stant we can define an alternative effective random walk
model by
?n??t,
t?n+1?= t?n?+ ?t.
?16?
?n??n=0
?
?n??n=0
?
?n?? a correlated Gaussian
L?t? in the variance of the
x?n+1?= x?n?+ ?x,
t?n+1?= t?n?+?x
vs
?n?.
?17?
For the heterogeneity scenarios under consideration, the
characteristic length is given by the correlation scale ? of the
log-hydraulic conductivity field. In fact, the correlation scale
of the ?Eulerian? flow velocity is in general larger than ? so
that a choice of ?x of the order of or smaller than ? should
be sufficient, i.e., ?x??. Here, the stochastic process
?vs
is defined by the Lagrangian ensemble statistics
along the projected particle trajectory at equidistant spatial
locations. In the following, we focus on the analysis and
characterization of the processes ?vt
?n??n=0
?
?n??n=0
?
and ?vs
?n??n=0
?.
III. NUMERICAL SETUP AND TRANSPORT BEHAVIOR
Here we consider different hydraulic conductivity fields
that have the same point distributions but different spatial
structures. Figure 1 illustrates three types of velocity fields
and the corresponding flow fields. We use particle tracking
simulations in d=2 dimensional heterogeneous hydraulic
conductivity fields in order to compute the Lagrangian ve-
locity and investigate its transition properties in space and
time.
We solve flow and transport in two-dimensional heteroge-
neous hydraulic conductivity fields. At the boundaries we
impose permeameterlike conditions: no flux across the lateral
boundaries and constant head at the upstream and down-
stream boundaries. The flow equation is solved numerically
with a finite difference scheme using a grid of 512?512
elements. Once the flow field is obtained, solute transport is
simulated by random walk particle tracking ?e.g., ?21??, i.e.,
numerical iteration of Eq. ?14?. Flow velocities are interpo-
lated using a bilinear interpolation scheme ?e.g., ?22??.
We consider here advection-dominated transport sce-
narios. The relative importance of advective and diffusive
transport mechanisms is measured by the Peclet number. The
latter is defined in terms of the geometric mean of the hy-
draulic conductivity field K¯, the pixel size d, and the diffu-
sion coefficient D as Pe=dK¯
advection-dominated and vice versa. The mean hydraulic
gradient is set to one. The Peclet number is set to Pe=102,
which is a typical value for solute transport in heterogeneous
geological formations. The influence of the Peclet number on
the velocity correlation properties will be investigated in a
further study. Solute particles are tracked within 100 hydrau-
lic conductivity field realizations. The particles are injected
along a centered vertical line of 200 elements length located
in a distance of 64 elements from the upstream boundary.
They are injected proportionally to the local flow. The nu-
merical solution of the flow problem and the implementation
of the particle tracking method are described in more details
in ?13?.
We consider three types of hydraulic conductivity fields:
?i? multilognormal fields, ?ii? connected multilognormal
fields, and ?iii? multilognormal stratified fields. For the mul-
tilognormal hydraulic conductivity field ?Fig. 1?a??, the vari-
ance is and the correlation length is ?=8 ?in units of ele-
ments?, i.e., the medium size is 64?64 ?. For the generation
of the lognormal hydraulic conductivity fields with con-
nected high hydraulic conductivity zones ?Fig. 1?b??, we use
the method described in Refs. ?23,24?. The resulting hydrau-
lic conductivity has the same point value distribution and
approximately the same two-point correlation as the multi-
lognormal hydraulic conductivity field described above. For
the stratified hydraulic conductivity field ?Fig. 1?c??, a hy-
draulic conductivity is randomly assigned to each ?one ele-
ment wide? horizontal stratum. Thus, here the correlation
structure is highly anisotropic as the longitudinal correlation
length is infinite, the transverse correlation length is of the
order of one.
The three hydraulic conductivity fields have identical
point distributions of hydraulic conductivity values. They are
different with respect to their respective spatial organization.
The degree of organization of the hydraulic conductivity
fields increases from top to bottom in Fig. 1. The calculated
point velocity distributions are given in Fig. 2. They are
D. For large Pe, transport is
SPATIAL MARKOV PROCESSES FOR MODELING…
PHYSICAL REVIEW E 78, 026308 ?2008?
026308-3

Page 4

highly heterogeneous with a range of about eight orders of
magnitude. The characteristic distance of the fields of the
first and second type is the correlation length ?, which, to-
gether with the geometric mean of the hydraulic conductivity
field K¯defines the characteristic advection time scale ?
=?/K¯. Note, however, that the velocity field organization
implies a range of ?Lagrangian? velocity correlation lengths
and times ?13?. The high heterogeneity of the hydraulic con-
ductivity field leads to connected high velocity zones and
strong focusing of the streamlines ?Figs. 1?a? and 1?b??. In
the following, we propose a method to quantify the Lagrang-
ian correlation lengths and times.
First, however, let us illustrate the effect of connectivity
on solute dispersion. To this end, we first display arrival time
distributions at a distance of 100 elements from the inlet
?Fig. 3?. The first arrival time decreases and the late time
tailing increases with increasing connectivity. Notice that the
late time tails are due to low particle velocities. However,
scale
log K and log V
10
(a)
(b)
(c)(d)
(e)
(f)
10
FIG. 1. ?Color online? Hydraulic conductivity K and modulus of velocity v. ?a? Multilognormal hydraulic conductivity field ??f
=8, size 512?512 elements?, ?b? corresponding velocity field, ?c? connected hydraulic conductivity field, ?d? corresponding velocity field,
?e? stratified hydraulic conductivity field, and ?f? corresponding velocity field. These three fields have the same point distribution of hydraulic
conductivity values.
2=9, ?
LE BORGNE, DENTZ, AND CARRERA PHYSICAL REVIEW E 78, 026308 ?2008?
026308-4

Page 5

even though all three hydraulic conductivity fields have simi-
lar distributions of the low velocity values ?Fig. 2? they dis-
play different tails of the first arrival time distribution. It
needs to be concluded that the breakthrough curve tailing is
not entirely controlled by the point velocity distribution. It is
also controlled by the velocity correlation properties. Thus
the hydraulic conductivity organization and its correlation
properties have a strong impact on the non-Fickian transport
properties and the relevant effective equation.
IV. LAGRANGIAN VELOCITY STATISTICS
For the stratified case, the Lagrangian velocity statistics
are known ?e.g., ?7,12,25??. The process is stationary and
characterized by a power-law correlation function. The latter
is due to the fact that velocity transitions here are equivalent
to particle transitions between strata by lateral diffusion.
Thus the probability to return to a given stratum, i.e., to a
certain velocity value decreases with travel time as t−1/2?e.g.,
?7,12??. The return probability and thus the Lagrangian cor-
relation is the same for all initial velocities. For the lognor-
mal hydraulic conductivity fields ?Figs. 1?a? and 1?b?? this is
different. As shown in the following, here the Lagrangian
velocity correlation at equidistant times, i.e., for the process
?vt
depends on the initial velocity vt
velocities turn out to be more strongly correlated than high
velocities.
In the following, we propose a methodology to character-
ize and analyze these processes in terms of transition prob-
abilities for the random processes ?vt
spectively. Recall that the former is defined by the
Lagrangian velocity statistics at equidistant times along the
particle trajectories while the latter is defined by the La-
grangian statistics at equidistant positions along the trajecto-
ries.
?n??n=0
?
?0?. Specifically, low
?n??n=0
?
and ?vs
?n??n=0
?, re-
A. Definition of velocity classes
We discretize the particle velocity into n−1 classes
?Ci?1?i?n−1of equal probability of occurrence. We define y
=P?v? as the score corresponding to the velocity v where
P?v? is the cumulative Lagrangian velocity distribution. We
discretize the y domain, which is bounded between 0 and 1,
into n−1 classes of equal width 1/?n−1?, defined by their
boundaries yi. The smallest velocity corresponds to y1=0 and
the largest velocity to yn=1. In this study, we use 49 classes
?n=50?. For a given velocity v the corresponding class Ciis
determined as follows: v?Ciif yi?P?v??yi+1.
B. Velocity transitions
We define t?x? as the first passage time of a particle at the
longitudinal location x. For a particle at longitudinal distance
x from the inlet, at time t?x?, we define the longitudinal La-
grangian velocity as
vL=
xn+1− x
t?xn+1? − t?x?,
?18?
where xn+1is the downstream coordinate of the pixel in
which the particle is moving. Thus the Lagrangian velocity is
defined as the average velocity across an incremental length
xn+1−x. It includes both advective and diffusive motions.
The practical advantage of this definition is that all velocities
vLare positive. Particle velocities along particle trajectories
are quantified at given times vt?t? and at given longitudinal
locations vs?x?.
We consider velocity transitions along the particle trajec-
tory ?i? depending on the travel time and ?ii? depending on
the travel distance along the trajectory projected on the mean
flow direction. Thus we define ?i? the probability for a par-
ticle to make a transition from velocity v? at travel time t? to
the velocity v at travel time t,
rt?v,t?v?,t?? = ????v − vt?t????vt?t??=v?
and ?ii? the probability for a particle to make a transition
from velocity v? at travel distance x? to the velocity v at
travel time x,
?19?
rs?v,x?v?,x?? = ????v − vs?x????vs?x??=v?.
?20?
Note that the transition distributions are stationary in time,
i.e., rt?v,t?v?,t??=rt?x,t−t??x?? and space, rs?v,x?v?,x??
=rs?v,x−x??v??, respectively, because the log-conductivity
f?x? is modeled as a stationary random field.
FIG. 3. Breakthrough curves for each of the hydraulic conduc-
tivity fields, at a distance equal to 100 elements from the particle
inlet. The breakthrough curves are averaged over 100 hydraulic
conductivity field realizations.
FIG. 2. Distributions of the decimal logarithm of absolute value
of the Eulerian velocity.
SPATIAL MARKOV PROCESSES FOR MODELING…
PHYSICAL REVIEW E 78, 026308 ?2008?
026308-5