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: tyagi@ifd.mavt.ethz.ch (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) 66966714 6713References

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$p1 $ kl1 ll j1

$pcj q: 56

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nl1krl=llF kri=liP F krik=liPnl1krl=llPl1j1$pcj krik=liPnl1krl=llPi1j1$pcjP ;i linl1

krlll

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F Xn

F l kXn krl $pl kXn krl $p1 kXn krl Xl1 $pcj

!51i li1

j1cj li

1 li j1cjj1

ituting Eq. (49) for pi in Eq. (47) leads to

F krik $ p Xi1

p

! krik $p krik

Xi1$p ; i 2 f1; . . . ; ng; 50pi p1 X

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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