-
Manav Tyagi a,*, Patrick Jenny a, Ivan Lunati b, Hamdi A.
Tchelepi c
ferent uid phases. The goal of this paper is to present the
general framework for this alternative modeling approach.
dened on the macroscopic (Darcy) scale. For slow single-phase ow
in a homogeneous porous medium,Darcys law is an expression of
momentum conservation at the macroscopic scale. When multiple
immiscible
* Corresponding author. Tel.: +41 44 632 4505; fax: +41 44 632
1147.E-mail address: [email protected] (M. Tyagi).
Available online at www.sciencedirect.com
Journal of Computational Physics 227 (2008) 66966714
www.elsevier.com/locate/jcp0021-9991/$ - see front matter 2008
Elsevier Inc. All rights reserved.Various one and two-dimensional
numerical experiments demonstrate that with appropriate stochastic
rules the particlesolutions are consistent with a standard
two-phase Darcy ow formulation. In the end, we demonstrate how to
modelnon-equilibrium phenomena within the stochastic particle
framework, which will be the main focus of the future work. 2008
Elsevier Inc. All rights reserved.
Keywords: Stochastic particle method; Lagrangian approach;
Porous media; Multi-phase ow; PDF method
1. Introduction
Flow and transport processes in natural porous media are usually
described using dierential equationsa Institute of Fluid Dynamics,
Sonnegstrasse 3, ETH Zurich, Zuerich, CH-8092,
SwitzerlandbLaboratory of Environmental Fluid Mechanics, GR A0 455,
Station 2, Lausanne, CH-1015, Switzerland
cDepartment of Energy Resources Engineering, Stanford
University, Stanford, CA 94305, USA
Received 1 March 2007; received in revised form 11 March 2008;
accepted 17 March 2008Available online 1 April 2008
Abstract
Many of the complex physical processes relevant for
compositional multi-phase ow in porous media are well under-stood
at the pore-scale level. In order to study CO2 storage in
sub-surface formations, however, it is not feasible to
performsimulations at these small scales directly and eective
models for multi-phase ow description at Darcy scale are
needed.Unfortunately, in many cases it is not clear how the
micro-scale knowledge can rigorously be translated into
consistentmacroscopic equations. Here, we present a new
methodology, which provides a link between Lagrangian statistics of
phaseparticle evolution and Darcy scale dynamics. Unlike in
nite-volume methods, the evolution of Lagrangian particles
rep-resenting small uid phase volumes is modeled. Each particle has
a state vector consisting of its position, velocity, uidphase
information and possibly other properties like phase composition.
While the particles are transported throughthe computational domain
according to their individual velocities, the properties are
modeled via stochastic processes hon-oring specied Lagrangian
statistics. Note that the conditional expectations of the particle
velocities are dierent for dif-A Lagrangian, stochastic modeling
frameworkfor multi-phase ow in porous
mediadoi:10.1016/j.jcp.2008.03.030
-
M. Tyagi et al. / Journal of Computational Physics 227 (2008)
66966714 6697uid phases are present (e.g., oil, super-critical CO2
and water), the permeability in Darcys original equationis replaced
by an eective value to accommodate the presence of other phases in
the porous medium [1]. Thiseective parameter is expressed as a
function of phase saturation and is called relative permeability.
Macro-scopic capillary eects are introduced by considering dierent
pressures in the dierent uid phases. The cap-illary pressure
relations are also usually expressed as functions of
saturation.
In addition to saturation, the relative permeability and
capillary pressure relations depend on the pore-scalegeometry,
network topology, wettability characteristics, viscosity ratio of
the uids, and saturation history.The physical interactions that
take place in the rock-uids system at the pore (microscopic) scale
dictatethe behavior observed at the macroscopic scale. The
complexity of the small-scale dynamics has precludedthe development
of a general approach that links the pore-scale physics and the
representation at the Darcyscale [2]. Thus, in practice the
relative permeability and capillary relations, which are assumed to
be appropri-ate macroscopic-scale descriptions, are obtained by
performing laboratory ow experiments using specimens(cores) of the
porous medium of interest.
This simple Darcy scale representation of the relative
permeability and capillary relations is thought to beapplicable for
two-phase ow under strongly wetting conditions when the viscosity
ratio is close to unity, andthe macroscopic ow is within a
relatively small range of capillary numbers [3]. In other cases of
practicalinterest, such as EOR (enhanced oil recovery) gas
injection processes and the injection and post injection peri-ods
associated with CO2 sequestration in aquifers and reservoirs, the
application of this simple model isquestionable.
The mean ow velocity of reservoir displacement processes is
quite small (a few centimeters per day) andthe characteristic pore
size of the medium is also very small. So the Reynolds number is
much less than unity,and the ow at the pore scale is expected to
usually be in the Stokes regime, in which the inertial eects
arenegligible and the pressure drop takes place entirely due to
viscous and capillary forces. In these cases, theproblem at the
pore level is well dened and can be solved, if the pore scale
geometry is known. However,even a small sample of a real porous
medium contains millions of pores and in most cases it is very
dicultto obtain the complete description of the pore scale geometry
[4].
While the small scale ow dynamics are interesting, the objective
is to construct a model based on rela-tions that represent the
macroscopic (Darcy scale and larger) behaviors accurately. The
model must accountfor the dynamic eects of the pore scale physics
on the large-scale ow. In the standard approach, theassumption is
that the pore scale physics is accounted for in the relative
permeability and capillary relations,which are obtained from
experiments. However, this standard treatment is not well suited,
if the owinvolves complex processes such as non-equilibrium
phenomena and residual trapping. In such ows, a sta-tistical
approach is more appropriate, since a small elemental volume of the
porous media contains a largenumber of pores. Here we develop a
statistical method for multi-phase transport in porous media using
sto-chastic particles.
Particle tracking methods have been employed successfully in
subsurface ow simulations. From the pio-neering works of [5,6],
fully Lagrangian schemes based on random walk approach have been
widely employedfor tracer (i.e. unit-viscosity, miscible,
single-phase) transport. Extension of the particle-tracking
approach tomore complex geometry [7] and reactive ows in highly
heterogeneous formations [8] appeared later. A
hybridEulerianLagrangian method, where particle tracking is
employed to represent the transport, was developedand used to model
unstable rst-contact miscible (two-component, single-phase ow)
displacements in thepresence of density and viscosity dierences
[9]. In these particle tracking methods, each particle representsa
physical mass. The concentration of the tracked species (e.g.
tracer) is obtained by averaging over the controlvolume. Relatively
large particle numbers and ne grids are necessary to obtain
reasonably accurate concen-tration distributions in the domain.
Several EulerianLagrangian schemes have been introduced for
linear tracer transport (see, e.g. [1014])and extended to nonlinear
problems such as solving the saturation equation for two-phase
immiscible ow(see, e.g., [1517]). Fully Lagrangian methods have
also been applied to reactive-tracer transport with nonlin-ear
accumulation term (see, e.g., [18,19]; or [20] for a comprehensive
review), which requires the calculation ofconcentration at the node
of a superimposed grid [21]. Unlike particle tracking schemes, here
concentrationsare propagated along path lines. Streamline-based
methods, which belong to this family, have been developed
for modeling multi-component multi-phase displacement processes
in heterogeneous domains [22]. Character-
-
istic bwith
behavA
to solve nonlinear hyperbolic problems with shocks numerically
using stochastic particles. Solving a nonlinear
nitely many particles, the size of the averaging volume can be
chosen innitely small, which allows to solvepurely hyperbolic
problems. Note, however, that most macroscopic physical scenarios
of interest depict dif-
6698 M. Tyagi et al. / Journal of Computational Physics 227
(2008) 66966714fusive eects; e.g. due to capillary pressure
dierences or pore scale dispersion.Note that it is not intended to
employ SPM to solve problems, which can already be computed with
con-
tinuum methods. The motivation is a framework, which oers an
alternative modeling approach, i.e. from aLagrangian viewpoint.
Such a Lagrangian framework is a natural way to represent
non-equilibrium phenom-ena by specifying the physical rules
governing the particle evolution at the micro-scale. Moreover, a
consistentprobability density function (PDF) transport equation can
be formulated, which allows to derive correspond-ing Eulerian
moment equations [26].
We demonstrate that in the limiting case of zero correlation
time and length scales, the macroscopic equa-tions derived from the
microscopic model reduce to the standard Darcy scale (macroscopic)
equations. Inmore general cases, however, additional terms and
closure models are required (which requires no modelingin the
stochastic particle method), if an Eulerian approach is used. There
are no inherent limitations in themethodology, provided the
required Lagrangian statistics can be specied, e.g. from
experiments or pore net-work simulations. Such a consistent
multi-scale multi-physics framework allows for more insight into
thephysics governing multi-phase ow in natural porous media;
moreover, this framework can help in derivingeective coecients and
proposing modied macroscopic models.
2. Basic ideas
In this section, we explain the basic ideas of the stochastic
particle method (SPM). Therefore, the nonlineartransport
problem
oqiot
r F i qi for i 2 f1; . . . ; ng on X 1
with some boundary conditions at oX is considered, where qi, F i
and qi are the density of a conserved scalar,ux vector and rate of
production, respectively. Now, we consider a large number of
computational particles,each associated with one of the n scalars.
Here, the density qi wiqpni represents the concentration of
i-parti-cles, where wi is the particle weight and q
pni the particle number density. Next, it is shown how to evolve
the
particles in order to compute the solution of Eq. (1). By
integrating Eq. (1) over a control volume X0 X oneobtainsZ
0
oqiot
dXZ
0r F i dX
Z0qi dX for i 2 f1; . . . ; ng: 2hyperbolic problem using
stochastic particles requires the estimation of ensemble averaged
quantities (e.g. sat-uration) which implies averaging over nite
volumes. For numerical reasons this does not work for
discontin-uous solutions and therefore, a minimum amount of diusion
(depending upon the size of averaging volume)must be introduced.
Note that the size of the averaging volume determines the
resolution. In the limit of in-ior at the pore scale.similar
approach was proposed in a previous, unpublished attempt [25]. It
was found that, it is impossibleWe developed a stochastic particle
based model for nonlinear immiscible multi-phase ow, where thephase
ux is a nonlinear function of saturation (e.g., as for the
BuckleyLeverett problem). In ourapproach, a particle belongs to a
specic uid phase (e.g., water and oil particles) and moves with the
phaseparticle velocity. The saturation is a statistical quantity
dened for an ensemble of the particles. Thus, ourmethod is dierent
from the characteristic based methods, where particles move with
the characteristicvelocity and saturation is associated with the
particles. In the stochastic particle method (SPM) framework,we
essentially construct a model for the large scale dynamics based on
stochastic rules for the phase particleased methods have been
employed for nonlinear immiscible two-phase ow where particles are
movedthe characteristic velocities and saturation is a particle
property [23,24].X X X
-
where
handcentr
The sticle,
which
where
are p(V i
It shoTheseume o
M. Tyagi et al. / Journal of Computational Physics 227 (2008)
66966714 6699the domain, particles might be removed or additional
particles are introduced; according to the local massPnow/Si
uld be noted that at this point the source terms qi represent
only well rates and are explicitly specied.source terms can easily
be treated in the SPM framework. Each particle represents either
mass or vol-f the uid phase to which it belongs. Therefore, to be
consistent with the source term at some location in/oSiot
$ F i qi; i 2 f1; . . . ; ng; 8which is consistent with Eq. (1).
In accordance with Eq. (4), each particle moves with velocity
ui F i
: 9roportional to the particle number density. Assuming that the
uids and the rock are incompressibleconstant and / /x) we can write
the saturation equations asX Xwhere the porosity is dened as /
Pnj1qj. The phase saturations
Si qi/ V iq
pni
/7ni X0qi dX
X0 V idX; 6R
0 qi dX is the volume represented by these particles. The
accessible pore space inside X0 is
R0 /dX,evolved according to Eq. (4). Moreover, since this is
true for any arbitrary volume X0 X, the particle solu-tion
converges to the exact solution of Eq. (1) for jX0j ! 0 and an
innite number of particles.
3. Model for multi-phase ow in porous media
In this section, it is shown how the SPM introduced in the
previous section can be employed to solve formulti-phase transport
in porous media. We consider n phases, each represented by a number
of computationalparticles. All particles of phase i have the same
mathematical weight wi V i, where V i is the volume occupiedby a
particle. One could also choose wi being proportional to the mass
associated to a particle and the volumebeing a function of pressure
and weight. For incompressible uid it is easier, however, to
directly take the par-ticle weight to be equal to the volume. In an
arbitrary volume X0 X, the number of phase i particles is equal
to
X0Z
pnZ
qiqi ui mdC qi qi mdC
wi mdC; 5
is identical to dF i in Eq. (3). Thus, we have shown that Eq.
(2) is solved consistently, if the particles arei i i
particle locations. The rate of particles owing across a surface
element dC is
pn pn F i F iuperscript denotes that a quantity is a particle
property. While ui is the velocity of an individual par-qpn is the
particle number density in its neighborhood. Note that we assume to
know both F and q at theui F iqi F i
wiqpni: 4side (rate at which particles are created inside X0).
Now, we show that the evolution of the particle con-ation qi is
consistent with Eq. (1), if the i-particles are transported with
the velocitynumber ni of i-particles in X changes. The rst
right-hand side term describes the contribution due to particleuxes
across the boundary oX0, which has to balance the left-hand side
term and the last term on the right-ni
m is the unit normal vector at oX0 pointing outwards. The rst
term in Eq. (3) is the rate at which theX0 0Using Gauss theorem and
with the relation qi wiqpni one obtains the equivalent
expressionoot
ZX0qpni dX|{z}X0
ZoX0
F iwi mdC|{z}dF i
ZX0
qiwidX for i 2 f1; . . . ; ng; 3rate. Summing Eq. (8) over all
phases and using the fact that j1Sj 1 we obtain
-
particles move with the velocity given by Eq. (9). Here, for
illustration purpose, it is assumed that the large
whertions of saturations, and therefore, the elliptic pressure
equation (see Appendix for the derivation)
is coupute
In gethe v
to bewithaveraging volume such that ergodicity can be assumed,
which allows to replace ensemble by spatial averaging.
IncontiTherequate
6700 M. Tyagi et al. / Journal of Computational Physics 227
(2008) 66966714real porous media ows there are various pore scale
phenomena that result in dispersive eects at thenuum scales, e.g.
due to capillary pressure dierences, molecular diusion and
mechanical dispersion.fore, there exist no innitely sharp fronts at
the macroscale. Mechanical dispersion can be treated ade-Below we
discuss two ways how this can be achieved.
3.1. Random walk methodextremely small and a huge number of
particles have to employed. However, in a numerical simulationa
nite size averaging volume, one has to ensure that the particle
distribution is nearly uniform over thein its neighborhood. For
smooth saturation distributions this can be achieved by averaging
over an ensemblearound the particle location x and since for an
innite number of particles the volume containing that ensem-ble can
be chosen innitely small, this local spatial averaging procedure
becomes identical with ensemble aver-aging at the location x. If
one insists in computing very sharp saturation fronts, then the
averaging volume has/oSiot
$ F i qi; i 2 f1; . . . ; ng: 15
neral, the phase uxes F i are functions of saturations and their
gradients. Therefore, in order to computeelocity (9) of a particle,
the saturations, i.e. the phase particle number densities qpni ,
have to be estimatedtransport equationspled with the phase
transport equations (8). One possible way to solve the system of
equations is to com-the pressure eld at the beginning of each time
step by solving Eq. (14) and subsequent solution of the$ kk$p1 $
kXn
l1 klXl1
j1$pcj
q 14e ki kri=li are the phase mobilities and k Pn
j1kj the total mobility. In the general case, ki are func-In
Appendix, we show how Eq. (11) can be rewritten in the following
fractional ow formulation
F i kik F kik
Pnl1kl
Pl1j1$pcj kikk
Pi1j1$pcj
k; i 2 f1; . . . ; ng; 13$ j1 lj
$pj q: 12scale (Darcy scale) uxes F i are governed by Darcys law
and read
F i krikli$pi; i 2 f1; . . . ; ng; 11
where k is the rock permeability and kri , li and pi are the
relative permeability, viscosity and pressure of phasei,
respectively. Usually, empirical expressions are used to relate kri
and pci1 pi1 pi with the phase satura-tions. Substituting the uxes
(11) into Eq. (10) leads to the following elliptic equation
Xn krjk !z}|{F$ Xn
j1F jz}|{F
Xn
j1qj
z}|{q: 10
In the absence of sources and sinks, the total ux F is
divergence free.So far, we have derived a discrete representation
that is consistent with the continuum Eq. (8), where phasely by
adding a diusion term to Eq. (8), which leads to the modied
saturation equation
-
Notenumescale
If one is not interested in resolving the length scales
associated with physical diusion, an alternative
ond o
i
wherevectoment
M. Tyagi et al. / Journal of Computational Physics 227 (2008)
66966714 6701the total ux F is evaluated at x and all other
quantities at x . Note that the components of ther n n1; n2; n3T
are independent random variables with standard Gaussian
distribution. It should beAll terms in the brackets are evaluated
at location xn. Finally, the new particle position is obtained
through
xn1 xn F i
/Si 1/$D
dt
2Ddt/
sn; 19
n1=2 nrder scheme a particle is rst transported according to
xn1=2 xn F i
/S 1/$D
dt2: 18approach can be used. First particles are moved with the
velocity given by Eq. (9). After each time stepone assigns
arbitrary new particle positions within the same cell.
Computationally, this can be done by uni-formly redistributing
(shaking) the particles in a cell after each time step. With a
uniform distribution ofparticles in a cell, local ergodicity can be
assumed, i.e. the particles in a cell represent the distribution
atone point in space and time. Important is that thereby the other
particle properties are not aected, such thatthe particle
properties of the ensemble still exhibit the same distribution.
Note, however, that the resulting dif-fusion is dictated by the
grid and cannot directly be controlled.
4. Solution algorithm
We solve the ow equation (14) for pressure using a nite-volume
method (FVM). The macroscopic trans-port Eq. (16), on the other
hand, is solved with the SPM, which is fully Lagrangian method,
where the particleevolution is given by Eq. (17). To solve the
transport equation, the whole domain is populated with variousphase
particles (consistent with the initial condition). The following
quantities are needed to evolve the par-ticles: F, Si, $pc and $D.
Since these quantities are only available at the grid level, they
have to be interpolatedto the particle locations. In order to
ensure mass balance, linear interpolation of F from the cell faces
to theparticle locations is used. To estimate Si, simple cell
averaging is employed at this point. The evolution of anindividual
particle is computed with a second or fourth order RungeKutta
scheme. For example, in the sec-/Si / /
where F i is given by Eq. (13). Note that F i also include
diusive uxes due to capillary pressure dierences.The second
right-hand side term of Eq. (17) is a Wiener process, where each
component ofdW dW 1; dW 2; dW 3T has a Gaussian distribution with
hdW ki 0 and hdW k dW li dkl dt. The last termis necessary to
account for spatially varying coecient D. One should remember that
Eq. (17) does not ac-count for the source term in Eq. (16). As
mentioned before, in order to account for the source term,
particleshave to be introduced or destroyed at consistent rates. In
our simulations, this is necessary in grid cells, whichare
perforated by a well.
3.2. Shaking methoddxi ui dt F i
dt 2D
sdW 1 $Ddt; 17size of the averaging volume must be in the
neighborhood of l. A particle evolution in physical space, which
isconsistent with Eq. (16), is given bythat in general D is not
constant in space and depends on jF ij and on the saturation. To
compute arical solution in the presence of these dispersive
phenomena we only need to resolve the smallest lengthl that
captures the corresponding eects at the continuum scales. In
practice, therefore, the characteristic/oSiot
$ F i $ D$Si qi; i 2 f1; . . . ; ng: 16ioned that in the above
scheme, higher order accuracy is obtained only if Si and D in Eq.
(19) are
-
5. Nu
1 1
tities
Fo
D
6702 M. Tyagi et al. / Journal of Computational Physics 227
(2008) 66966714is one, where Dx is the grid spacing used for the
SPM. Note that the SPM requires a grid Peclet number, whichis not
much larger than one. Figs. 1 (a) and (b) show the saturation
proles of phase two for M 1 after thetime t0 0:25 obtained with the
grid spacings Dx0 0:01 and 0.002, respectively and Pe 1. The good
agree-ment between the SPM (solid lines) and FVM (dotted lines)
solutions demonstrates that the two methods areconsistent. Note
that the grid spacing used for the FVM is Dx0=10 in order to
provide a good reference. Inorder to keep the statistical error of
the SPM results very small, a huge number of particles, i.e. on
average50,000 per grid cell, were employed. Note, however, that the
SPM also works with much fewer particles.Fig. 2 shows SPM and FVM
results for various viscosity ratios M and a grid spacing Dx0 0:01
(0.001 forsionless space and time coordinates x x=L and t Ft=/L are
used. At time t 0, the saturationS1 S is one in the whole
reservoir. Then, for t0 > 0, phase two is injected at the left
boundary(S2 1 S 1 at x0 0). The total volume ux F is constant
during the whole simulation. For the rststudies, no capillary
pressure eect is considered, i.e. pc 0, and the diusion coecient D
is chosen suchthat the grid Peclet number
Pe FDx 23the Fr initial validation purpose, a simple 1D problem
is considered. In the following results, the dimen-0 0 05.1. 1D
validationare presented in dimensionless form.where M l2=l1 is the
viscosity ratio. The saturations of phase one and phase two are S1
S and S2 1 S,respectively. To treat stochastic component of diusion
we employ random walk method. Note that all quan-k1 S2
land k2 1 S
2
Ml; 22pc p1 p2 21is the capillary pressure dierence between the
two phases. We take the quadratic relative permeabilities
func-tions asIn this section we demonstrate that the SPM is
consistent with the governing equations. In order to
showconvergence, the SPM results are compared with corresponding
FVM reference solutions (using up to tentimes ner grids to avoid
numerical dispersion). For all the results presented here two
phases (n 2) withthe ux functions
F1 k1k F kk1k2k
$pc and F2 k2kF kk1k2
k$pc 20
are considered, wheremerical validationevaluated at the mid
point xn1=2
. A more detailed discussion on higher order integration of
stochastic dier-ential equations can be found in [27]. Initially,
the particles are distributed according to the specied satura-tion
distribution. At the beginning of each time step, F is computed by
the FVM and subsequently theparticles are transported as described
above. In order to deal with in- and out-ow boundary conditions,the
computational domain is surrounded with a layer of ghost cells,
which can be re-populated consistentlywith the specied boundary
conditions at the beginning of each time step [28]. Re-population
is also appliedin cells where a source is employed (wells). No-ow
boundaries are treated by simply reecting the particles atthe
corresponding walls.VM).
-
M. Tyagi et al. / Journal of Computational Physics 227 (2008)
66966714 6703Ne
Subst
with
and Cthe pcan bminiminitiaFig. 1. Simulation results of the 1D
test case for M 1: (a) Dx 0:01; (b) Dx 0:002.0.1 0.2 0.3 0.4
0.5x
0
0.2
0.1 0.2 0.3 0.4 0.5x
0
0.2
0 00.4 0.40.6
S2
0.6
S20.8
1SPM solution (100 grid cells)FVM solution (1000 grid cells)
0.8
1SPM solution (500 grid cells)FVM solution (5000 grid cells)
a bxt, we consider the same 1D problem with capillary pressure,
i.e. with
pc p0=S for drainagep0=1 S for imbibition:
24
ituting Eqs. (22) and (24) into Eq. (20) leads to the ux
functions
F1 MS2
MS2 1 S2 F C$S and 25
F2 1 S2
MS2 1 S2 F C$S 26
C C0
1S2MS21S2 for drainage
C0 S2
MS21S2 for imbibition
8 0:05, Pe was increased
Fig. 2. Simulation results of the 1D test case for dierent
values of M 4; 2; 0:5; 0:25; 0:1 and Dx0 0:01.
-
by a factor of 10 and C was nite. Figs. 3 (a) and (b) show the
saturation proles at t0 0:25 for C0 0:01 andC0 0:02, respectively.
In all cases, the grid spacings for the SPM and the FVM are 0.01
and 0.001, respec-tively. It can be seen that the SPM (solid lines)
and FVM (dotted lines) results are in excellent agreement. Figs.4
(a) and (b) depict the corresponding results for imbibition.
5.2. 2D validation
The 1D validation studies show that the SPM is consistent with
the FVM and that the results converge tothe correct solutions.
Here, it is demonstrated that the method can also be applied for
multi-dimensional prob-lems. We consider a quadratic 2D domain
(quarter-ve-spot conguration) X of size L L with impermeablewalls
(Fig. 5). A source and a sink are distributed over square
sub-domains as
q 100L2q0; if 0 6 x=L 6 0:1 ^ 0 6 y=L 6 0:1
100L2q0; if 0:9 6 x=L 6 1 ^ 0:9 6 y=L 6 1
0; else;
8>: 28where phase two is injected at the lower left corner
(the viscosity ratio M is one in all cases). Initial conditionsat t
0 are
0.1 0.2 0.3 0.4 0.5x
0
0.2
0.4
0.6
0.8
1
S2
SPM solution (100 grid cells)FVM solution (1000 grid cells)
0.1 0.2 0.3 0.4 0.5x
0
0.2
0.4
0.6
0.8
1
S2
SPM solution (100 grid cells)FVM solution (1000 grid cells)
a b
6704 M. Tyagi et al. / Journal of Computational Physics 227
(2008) 669667140.1 0.2 0.3 0.4 0.5x
0
0.2
0.4
0.6
0.8
S2
0.1 0.2 0.3 0.4 0.5x
0
0.2
0.4
0.6
0.8
S21SPM solution (100 grid cells)FVM solution (1000 grid
cells)
1SPM solution (100 grid cells)FVM solution (1000 grid cells)
a bFig. 3. Simulation results with M 1 and a constant diusion
coecient: (a) C0 0:01; (b) C0 0:02.Fig. 4. Simulation results with
M 1 for imbibition: (a) C00 0:01; (b) C00 0:02.
-
5.2.1.
Fihomodispe
(0,L)
Y
wall
wall
wall
wallX(L,0)
(L, 0.9L)
(0.9L, L) (L, L)
(0,0) (0.1L,0)
(0, 0.1L)
Fig. 5. Quarter-ve-spot conguration. The shaded regions
represent the distributed source (left-bottom corner) and sink
(top-right
Fig. 6satura
M. Tyagi et al. / Journal of Computational Physics 227 (2008)
66966714 6705Homogeneous case
rst, in order to demonstrate that the SPM can be applied for
non-uniformmulti-dimensional simulations, aS2 1; if 0 6 x=L 6 0:11
^ 0 6 y=L 6 0:110; else:
29
The domain is discretized by an orthogonal grid into 100 100
cells of equal size and for all the followingresults the
dimensionless space and time coordinates x0 x=L and t0 q0t=/L2 are
used. With the followingstudies we want to demonstrate that the SPM
for transport gives consistent results. Although it is possible
toupdate the ow eld every time step, here the focus is on the
transport part and the ow was computed only atthe beginning of the
simulations. For all the following studies an average number of
16,000 particles per celland a fourth order particle tracking
scheme were employed. Moreover, for validation purpose the SPM
resultsare compared with the corresponding FVM solutions, for which
a QUICK scheme [29] was used.
corner).geneous permeability eld is considered. No capillary
pressure eects are taken into account (C 0) andrsion is purely
mechanical with D 0:01LjFj. This corresponds to a grid Peclet
number of one everywhere
0.05
0.05
0.05
0.6
0.6
0.6
0.7
0.7
0.7
x
y
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1SPM solutionFVM solution (QUICK scheme)
0.2 0.4 0.6 0.8 1diagonal
0
0.2
0.4
0.6
0.8
1
S2
SPM solutionFVM solution (QUICK scheme)
a b
. Simulation results for quarter-ve spot case in homogeneous
permeability eld and M 1: (a) contours of injected phasetion; (b)
variation of injected phase saturation along the diagonal.
-
Fig. 7. Particle distributions in homogeneous permeability eld
and M 1: (a) phase-2; (b) phase-1.
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
y
25
24
23
22
21
20
19
18
17
16
15
Fig. 8. Permeability eld for heterogeneous test case (log
k).
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.6
0.6
0.6
0.6
0.60.6
0.7
0.7
0.70.7
x
y
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1SPM solutionFVM solution (QUICK scheme)
Fig. 9. Simulation results for quarter-ve spot case with
heterogeneous permeability eld and M 1; shown are contours of the
injectedphase saturation.
6706 M. Tyagi et al. / Journal of Computational Physics 227
(2008) 66966714
-
in the domain. The time step, Dt0, was equal to 0.001
corresponding to a maximum CFL number of approxi-mately 0.5. Fig. 6
(a) depicts contours and Fig. 6 (b) proles (along the diagonal from
injector to producer)of S2 at t0 0:25. Shown are both, SPMandFVM
results and as can be observed they are in excellent agreement.In
addition, scatter plots of the phase two and phase one particles
are shown in Fig. 7 (a) and (b), respectively. Itshould be noted
that the sparsely distributed particles in Fig. 7 (b) represent the
expansion wave and are not dueto diusive eects.
5.2.2. Heterogeneous case
Here, a more realistic case with the heterogeneous permeability
eld depicted in Fig. 8 (log k) is considered.As in the previous
study, dispersion is solely due to mechanical dispersion, i.e. C 0
and D 0:01LjFj. Sincethe velocity variation is larger this time,
Dt0 was 0.0001 to ensure a CFL number smaller than one
everywhere.Again, excellent agreement between SPM and FVM results
can be observed in Fig. 9, where the contours of S2are shown. The
phase particle distributions are depicted in Fig. 10 (a) and
(b).
5.2.3. Homogeneous case with capillary pressure eects
Finally, convergence is demonstrated for a 2D case with
capillary pressure eects using the values 0.01 and0:001LjFj for C0
C=q0 and D, respectively. Without loss of generality a homogeneous
permeability eld was
M. Tyagi et al. / Journal of Computational Physics 227 (2008)
66966714 6707Fig. 10. Particle distribution in heterogeneous
permeability eld for M 1: (a) phase-2; (b) phase-1.
0.05
0.05
0.05
0.6
0.6
0.6
0.7
0.7
0.7
x
y
0.05
0.05
0.05
0.6
0.6
0.6
0.7
0.7
0.7
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1SPM solutionFVM solution (QUICK scheme)
0.2 0.4 0.6 0.8 1diagonal
0
0.2
0.4
0.6
0.8
1
S2
SPM solutionFVM solution (QUICK scheme)
a b
Fig. 11. Simulation results for quarter-ve spot case with
homogeneous permeability eld, capillary pressure (C0 0:01) and M 1:
(a)
contours of injected phase saturation; (b) variation of injected
phase saturation along the diagonal.
-
usedcases
6. Pr
In
systemary. T
particthat t
get th
for th
which leads to the conservation law
6708 M. Tyagi et al. / Journal of Computational Physics 227
(2008) 66966714oSaot
oox
S2aS2 S2
( ) 0 371 2
If the specic rule
L 1; with the probability Sax; t
0; else
35
is employed, hLja ai becomes equal Sa and as a result Eqs. (33)
and (34) can be rewritten as
u L
S21 S22; F 1 S
21
S21 S22and F 2 S
22
S21 S22; 36F 2 S2hLja 2i
S hLja 1i S hLja 2i : 34u S1hLja 1i S2hLja 2i 33
and the particle mass uxes can be expressed as
F 1 S1hLja 1i
S1hLja 1i S2hLja 2iande particle velocity:
LBy substituting for the pressure gradient in Eq. (30) using the
relations (31), one gets the following expressionF 1 F 2 1: 32F 1
S1hL ja 1i ox and F 2 S2hL ja 2i ox : 31
Due to continuity and the imposed inow conditions at the left
boundary one obtains the relatione particle mass uxes
op opwhere the property a a indicates the phase represented by
the particle. By multiplying the conditional par-ticle velocities
with the corresponding saturations Sa, which can be regarded as
particle number densities, wele moves with the velocity u op=ox
according to Darcys law for one-phase ow and L 0 meanshe particle
is immobile. The mean conditional particle velocity huja ai is
equal to hLja aiop=ox,
where op=ox is the macroscopic pressure gradient and L a
particle mobility. In other words, if L is one, theu L opoxx; t;
30particle entering through the left boundary is u 1. We impose the
rule that a particle representing phasea 2 f1; 2g moves with the
velocity, initially occupied by phase one, where phase two
particles are entering the domain at the left bound-he condition at
the left boundary is specied as constant total ux F 1, so that the
velocity of eachsimple statistical rules describing the evolution
of particle properties. Note that the aim is not to propose a
newphysical model; the scenario described next is of illustrative
nature only. We consider an incompressible 1Dobability density
function (PDF) modeling of non-equilibrium multi-phase systems
this section, we demonstrate how the SPM framework can be used
to derive macroscopic behavior fromfor this study. Fig. 11 (a)
depicts contours and Fig. 11 b diagonal proles of S2. As in the
previous 2D test, the agreement between the SPM and FVM solutions
is excellent.1 2
-
for Swith qscale
for thsoluti
Ne
dW, wdepenstruct
No
whichand Lthe JP
tiplyitrans
whichcan d
M. Tyagi et al. / Journal of Computational Physics 227 (2008)
66966714 6709whole sample space then leads to
oSahLja^ aiot
oox
SahLja^ ai2S1hLja^ 1i S2hLja^ 2i
( ) oox
SahL hLja^ ai2ja^ aiS1hLja^ 1i S2hLja^ 2i
( ) x hLja^ ai L 0: 46is identical with (37) for hLja^ ai Sa
showing consistency with the equilibrium model. Similarly weerive a
transport equation for SahLja^ ai by multiplying Eq. (44) with bL2
a^. Integration over the
oSaot
oox
SahLja^ aiS1hLja^ 1i S2hLja^ 2i
0; 45ng the JPDF Eq. (44) with 2 a^ and subsequent integration
over the a^bL-space leads to the saturationport equationdt f obLin
the x-;a^- and bL-directions and therefore the modeled JPDF
equation becomes
ofot oox
bLfS1hLja^ 1i S2hLja^ 2i
( ) oobL xa^bL La^fn o
o2r2a^xa^f obL2 0: 44
This is a FokkerPlanck equation, where the rst term describes
the temporal change of the f, the second termits transport in
physical space, and the third and fourth terms describe drift and
diusion in the bL-space. Mul-dt
dL ja^; bL; x; t xa^ bL La^ 1 or2a^xa^f 43dXdt
ja^; bL; x; t bLS1hLja^ 1i S2hLja^ 2i ; 41
da ja^; bL; x; t 0 and 42w we write the general form of the
joint probability density function (JPDF) evolution equation,
i.e.
ofot oh
dXdt ja^; bL; x; tif
ox oh
dadt ja^; bL; x; tif
oa^ oh
dLdt ja^; bL; x; tif
obL 0; 40can be derived from the conservation law for the joint
probability density f of the stochastic variables a. Note that a^
and bL are the corresponding sample space coordinates,
respectively. In our specic model,DF is transported with the
velocities hLja ai La and hL hLja ai2ja ai r2a.hich follows a
Gaussian distribution with hdW i 0 and hdW 2i dt. Note that the
diusion coecientds on the equilibrium variance r2a and on the rate
xa. The stochastic dierential equation (39) is con-ed such that for
constant coecients L reaches a Gaussian equilibrium distribution
withdL xa L La
dt 2r2axa
qdW 39
for the evolution of L of a phase particles. The rst term on the
right-hand side describes the relaxation of L
to some equilibrium value La at the rate xa. The last term is a
stochastic diusion term with the Wiener processe phase a particle
velocities directly in terms of the saturation values and obtains
the same macroscopicons.xt, we introduce non-equilibrium eects by
considering the Langevin modelto Sa independent of the particle
property PDF and one can also write
u SaS21 S22
38a. Note that Eq. (37) is identical with the standard two-phase
Darcy formulation of incompressible owuadratic relative
permeabilities, constant viscosities, unit porosity and without
capillary pressure or poredispersion. In this simple case of a
system in equilibrium, the conditional expectation hLja ai is
equala a
-
Note that the moment Eqs. (45) and (46) do not form a closed
system, since in general the third term on theleft-hand side of Eq.
(46) is unknown. On the other hand, this closure problem is avoided
by directly solvingthe JPDF Eqs. (44), e.g. with the SPM.
The example above simply demonstrates how the SPM can be
employed to use statistical moments and cor-relation structures of
phase particle velocities (i.e. La, r2a and xa) to predict
consistent statistical macroscopicbehavior. The full advantage of
such a stochastic approach becomes apparent, for example, if
non-equilibriumphysics involving non-trivial PDFs is
considered.
6.1. Numerical results
Here we present a few simulation results, based on the
non-equilibrium model explained above. As alreadymentioned in the
previous subsection, the aim is to demonstrate how such simple
Lagrangian rules lead to theresults distinctly dierent from the
corresponding equilibrium Darcy solutions. Note that it is not
intended topropose a new physical model, however. For the studies
we consider the same one-dimensional test case as inSection 5.1.
Initially, the computational domain is populated with phase 1
particles, which are in equilibrium,i.e. a 1 and L S1 1. From the
left boundary, phase 2 particles with a 2 and L S2 1 enter
thedomain. The relaxation time (dimensionless) s0a 1=x0a s0 is the
same for all particles and ra is set to zeroeverywhere. Note that
this model leads to L S1 and L S2 for phase 1 and phase 2
particles, respectively, if
0.4 0.8 1.2 1.6 20
0.2
0.4
0.6
0.8
1
S2
= 1= 0.1= 0
0.4 0.8 1.2 1.6 20
0.2
0.4
0.6
0.8
1
S2
= 1= 0.1= 0
a b
6710 M. Tyagi et al. / Journal of Computational Physics 227
(2008) 66966714x x
Fig. 12. Saturation evolution for dierent values of s0: (a) t0
0:5; (b) t0 1:5.
0.2 0.4 0.6 0.8 1S2
0
0.2
0.4
0.6
0.8
1
kreff
t=0.25t=0.5t=1t=1.5equil
0.2 0.4 0.6 0.8 1S2
0
0.2
0.4
0.6
0.8
1
kreff
t=0.25t=0.5t=1t=1.5equil
a bFig. 13. The relative permeability curves as function of
time: (a) s0 0:1; (b) s0 1.
-
M. Tyagi et al. / Journal of Computational Physics 227 (2008)
66966714 6711s0 ! 0. Otherwise, the distribution of L will be
distinctly dierent; as the saturation proles. For all simula-tions
the viscosity ratio M 1, the grid spacing Dx0 0:01, dt0 0:005 and
the average number of particlesper cell was 50,000. Fig. 12 (a) and
(b) depict the injected phase saturation proles at two dierent
timesfor s0 0, s0 0:1 and s0 1. A signicant departure from
equilibrium can be observed. Figs. 13 (a) and(b) depict the eective
relative permeability, kreff S2hLja^ 2i, curves as function of
injected phase saturationat four dierent times. One can observe
that at late times the relative permeability curves approach a
self-sim-ilar prole, which is dierent from the equilibrium curve
and depends on s0.
7. Discussion
The numerical examples and comparisons of Section 5 demonstrate
that the SPM with appropriate rules forthe phase particle movement
is consistent with standard two-phase Darcy ow. In Section 6, it is
shown hownon-equilibrium eects can be modelled in the SPM
framework. It has to be emphasized, however, that thesestudies only
serve as a proof of concept for the SPM and demonstrate the power
of SPM in modeling complexnon-equilibrium phenomena. Below, the
implications for physical modeling, but also the numerical
dicultiesand challenges are further discussed.
7.1. Implications for physical modeling
The motivation for the development of such a SPM is a
computational framework, in which the histories ofindividual
(innitesimal) uid volumes can be modeled depending on their phase,
composition and other prop-erties. It is important to distinguish
between the SPM and deterministic particle methods such as
characteristicmethods or smooth particle hydrodynamics, where the
particles carry saturation values. A particle in the SPMrepresents
a uid phase and moves with the phase particle velocity as opposed
to the characteristic basedmethod, where a particle moves with the
characteristic velocity. As in the physical world, saturation is
repre-sented as a local, spatial average of phase volume ratios,
i.e. saturation is a statistical quantity and not a
particleproperty. Algorithmically, the saturation is estimated with
local support (e.g. for the studies presented in thispaper as cell
averages). We expect that various complex physical processes can be
described more directly andnaturally than in a pure Eulerian
framework, in which not individual uid particle histories, but the
evolutionof mean values (e.g. saturations) at x locations is
modeled. For example, as shown in Section 6, evolution of auid
particle, which depends on its history (and has memory) leads to
non-equilibrium eects. These play a cru-cial role for trapping,
dissolution and reaction processes and the SPM oers an alternative
approach to modelthem. Moreover, although the SPM is not a pore
scale model, it can provide a consistent link between the phys-ics
in the pores and the dynamics observed at Darcy scale. Therefore,
however, the Lagrangian statistics of uidparticles has to be
provided, e.g. from pore scale modeling. Note also that the
particle ensemble represents thejoint probability density function
(PDF) of the particle properties (and not only rst and maybe
secondmoments) as a function of space and time. Similar PDF methods
have been applied with considerable successto model turbulent
reactive ows [26]. There, they have the signicant advantages that
turbulent convectionand chemical reactions appear in closed form.
Moreover, the huge amount of statistical information containedin
the joint PDFs allows to develop more sophisticated models. On the
other hand it has to be emphasized that aSPM simulation requires
signicantly more computer resources than a FVM study of the same
test case. There-fore, we do not intend to use the SPM for very
large studies, but rather to investigate how the macroscopic(Darcy
scale) behavior relates to the physics and dynamics at the pore
scale. We hope that such insight will ulti-mately lead to improved
models for FVM simulators.
7.2. Numerical diculties and challenges
As mentioned above, the SPM is computationally much more
expensive than e.g. a FVM. This is due to thelarge number of
particles, which is required to keep the statistical and the
deterministic bias errors small. Forexample, the simulations
discussed in this paper employed 10,00050,000 particles per grid
cell. Unfortunately,the statistical error estat converges very
slowly, i.e. estat 1=
n
p, where n is the number of particles. Fig. 14depicts the eect
of n on the saturation prole. It is clear from the gure that even
with few particles the expec-
-
6712 M. Tyagi et al. / Journal of Computational Physics 227
(2008) 66966714tation of the solution is essentially the same, with
a large statistical error, however. To reduce the statisticalerror
without using too much memory, one can sample the results at
several independent simulations (usingdierent random number
generator seeds) and then average the individual results in a
postprocessing step. Notethat the individual simulations can be
done in parallel on standalone machines, i.e. no parallel computer
isrequired. However, this approach does not help to reduce the
deterministic bias error ebias [30]. In order to con-trol ebias, it
is important to pay attention that the number of particles per cell
is suciently high. Another issue,which is related to the required
particle number, is the estimation of statistical moments from the
particle eld.For example, to resolve sharp fronts, a high spatial
resolution is required. If the box method (sampling over gridcells)
is employed, this implies that a ner grid and therefore in total
more particles are required. The boxmethod is only of rst order
spatial accuracy and it is similar to the rst order nite volume
method. It is worth-while to investigate more sophisticated
techniques to estimate saturation locally at particle locations.
Possiblealternatives might be based on spectral or wavelet methods
[31]. In any case, however, one has to ensure that thedispersion in
SPM simulations is dominated by the physical model and not by
numerical errors.
8. Conclusions
In this paper, a stochastic particle method (SPM) to model
nonlinear transport in porous media ow is pre-sented. The
motivation is the development of a modeling framework, in which the
evolution of individual(innitesimal) uid volumes (particles) can be
modeled directly depending on their phase, composition, and
0.1 0.2 0.3 0.4 0.5x
0
0.2
0.4
0.6
0.8
1
S 2
n=100n=1000n=10000
Fig. 14. The eect of number of computational particles used in
the SPM simulation. Here n is the average number of particles per
cellduring the simulation.other properties. We believe that for
various complex physical processes such a Lagrangian
modelingapproach is very natural and certainly provides an
alternative viewpoint. The goal of the present work wasto develop
the basic SPM solution algorithm and to demonstrate that it is
consistent with standard two phaseDarcy ow, if appropriate rules
for the particle evolution are employed. Therefore, various one-
and two-dimensional validation studies were performed, which show
that the SPM results converge to the expectedsolutions. The
numerical algorithm requires a minimum amount of dispersion, i.e.
pure shocks cannot be com-puted accurately. The amount of
dispersion needed depends on the scheme used to estimate
statisticalmoments, which may be improved in the future.
Furthermore, with a simple illustrative example it is shownhow the
SPM can be used to model non-equilibrium transport eects. As a next
step, it is planned to demon-strate that the SPM provides an
alternative and attractive approach to model some of the complex
phenom-ena, which are relevant for CO2 storage and which have their
origin at the pore scales.
Acknowledgments
The authors wish to acknowledge Dr. Benjamin Rembold with whom
they had many useful discussions ontechnical issues during the
development of the SPM C++ code. This research was supported by the
GlobalClimate and Energy Project at Stanford University, USA.
-
F i il $pi; i 2 f1; . . . ; ng; 47Supp
sures
Subst
The t
from
Subst
or
The P
M. Tyagi et al. / Journal of Computational Physics 227 (2008)
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A Lagrangian, stochastic modeling framework for multi-phase flow
in porous mediaIntroductionBasic ideasModel for multi-phase flow in
porous mediaRandom walk methodShaking method
Solution algorithmNumerical validation1D validation2D
validationHomogeneous caseHeterogeneous caseHomogeneous case with
capillary pressure effects
Probability density function (PDF) modeling of non-equilibrium
multi-phase systemsNumerical results
DiscussionImplications for physical modelingNumerical
difficulties and challenges
ConclusionsAcknowledgmentsReferences