1-s2.0-S009457650900438X-main

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Acta Astronautica

Acta Astronautica 66 (2010) 844–863

0094-57

doi:10.1

� Cor

E-m

(N.N. Sm

journal homepage: www.elsevier.com/locate/actaastro

Three-dimensional convection and unstable displacement of viscousfluids from strongly encumbered space

N.N. Smirnov �, V.F. Nikitin, V.R. Dushin, Yu.G. Phylippov, V.A. Nerchenko

Moscow M.V. Lomonosov State University, Moscow 119992, Russia

a r t i c l e i n f o

Article history:

Received 15 July 2009

Accepted 30 August 2009Available online 26 September 2009

Keywords:

Convective flow

Displacement

Viscous fluid

Porous medium

Instability

65/$ - see front matter & 2009 Elsevier Ltd. A

016/j.actaastro.2009.08.028

responding author. Tel.: þ7495 9391190; fax:

ail addresses: [email protected],

irnov).

a b s t r a c t

Purpose of the present paper is to investigate 3D instability effects in convective flows of

viscous fluid displaced by a less viscous one from strongly encumbered space, and to

determine characteristics of displacement quality. Fluids are assumed incompressible

and miscible. Extensive direct numerical simulations are used to study the sensitivity of

the displacement process to variation of values of the main governing parameters.

Comparison with results of two-dimensional simulations enabled us to investigate the

effect of aspect ratio on instability growth in viscous fluids displacement. A 1D model

with two fitting parameters is created in order to simulate behavior of the cross section

averaged parameters of the flow.

& 2009 Elsevier Ltd. All rights reserved.

0. Introduction

The problem of fluid flows in essentially encumberedspace are relevant to fluid flows in heat exchangers ofdifferent types, wherein flow channels are blocked up bydifferent heat exchange elements. The present investigationis also relevant to studying the thermal fields establishedinside the spacecraft capsule under the flight conditions[1,2]. The great amount of the containers and complexity ofthe fluid–solid body interface is the main difficulty for themodel elaboration. The traditional approach requires thecalculation of fluid flow in a space of very complex formshaped containers and the boundary conditions should beformulated along the whole fluid–solid body interface. Toavoid this difficulty an original approach has been workedout performing calculation. The fluid flow in the capsule wassimulated using the model for fluid filtration in porousmedium of changeable permeability. The mathematicalmodel of fluid filtration through the medium with variablepermeability is more suitable for the numerical calculations

ll rights reserved.

þ7495 9394995.

[email protected]

as it does not need the formulation of great amount ofboundary conditions at the container surfaces and thecalculation of flow in the space of a rather complex formbetween the containers. To avoid these difficulties, it isenough to determine the medium in the containers as a lowpermeable and the medium between them as high perme-able [2]. Essentially different temperatures in fluid bring toviscosity variation, which in case of forced convectiondefinitely present under microgravity conditions createsthe situation of viscous fluid displacement by a less viscousone.

In frontal displacement of a more viscous fluid by a lessviscous one Saffman–Taylor instability of the interface couldresult in formation of ‘‘fingers’’ of displacing fluid penetrat-ing the bulk of the displaced one. The growth of fingers andtheir further coalescence could not be described by a linearanalysis. Growth of fingers causes irregularity of the mixingzone thus affecting the displacement quality and heatexchange forecasts.

The problems of seepage flows were studied by manyauthors [3–12]. Investigating instability in miscible displa-cement differs greatly from that in immiscible fluids. Thepresence of a small parameter incorporating surface tensionfor immiscible fluids allows to determine theoretically thecharacteristic shape and width of viscous fingers [7,8], while

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N.N. Smirnov et al. / Acta Astronautica 66 (2010) 844–863 845

in miscible fluids theoretical analysis allows to forecast theshape of the tips, but does not allow to determine the widthof fingers, which remains a free parameter [5,6]. Numericalsimulations of viscous fingering in miscible and immiscibledisplacement were carried out in [9,10]. Those paperscontain an extensive bibliography on the history of theresearch as well. Numerical simulations [11] made itpossible to explain new experimental results on the pear-shape of fingers and periodical separation of their tipelements from the main body of displacing fluid. Thoseseparated blobs of less viscous fluid move much faster thanthe mean flow of the displaced viscous fluid [12].

The results of numerical simulations allowed tointroduce dimensionless parameters characterizing thequality of displacement and the mixing flux induced byinstability [12]. In the paper [13] the asymptotic behaviorof miscible displacements in porous media was studied inthe two limits, where a permeability-modified aspectratio, became large or small, respectively.

The influence of inhomogeneity of porous matrix ondisplacement instability was investigated [13–15]. Themodified Hele–Shaw cell containing regular and rando-mized obstacles was used to model and study the effect ofinhomogeneity on displacement instability [12,14]. Re-sults of numerical simulations as well as physicalexperiment showed that the presence of inhomogeneityof a definite length scale could stabilize unstable dis-placement and could destabilize a stable one [14].

Most of viscous fingering numerical simulations wereperformed for two-dimensional problems; one of the firstclassical publications was by Homsy [9,16]. This workused the spectral methods approach. In reality displace-ment and induced instability have a three dimensionalnature. Papers [17–22] investigate 2D and 3D miscibledisplacement of fluids with account for gravity using thespectral methods, as in earlier works by Homsy.

First attempts to perform comparative analysis for theinstabilities arising in displacement from 3D cells ofdifferent aspect ratios were performed in [23,36].

The present paper is aimed at numerical investigationof incompressible miscible fluids displacement in 3Dgeometry porous medium and studying the displacementscenarios being functions of aspect ratios and otherdimensionless governing parameters.

For practical applications, such as simulating generalheat exchange, often it is not important to have a detailedpicture of viscous fingers development, rather then tohave a quantitative estimate of the mixing flux induceddue to displacement instability. Thus it is necessary toelaborate methods making it possible to simulate in-stability induced mixing within some integral approach,not sensitive to spatial resolution. A 1D model describingdynamic behavior of cross section averaged parameters isbuilt with two fit parameters. Developing those modelparameters for description of displacement quality andthe mixing flux due to instability is also one of the goals ofthe present research. The regular porosity and perme-ability variation as shown in [11,12] could stabilizedisplacement (reduce the disturbances growth rate),which is important for stable operation of heat exchan-gers.

1. The problem statement

1.1. Physical description

The parallelepiped domain filled with porous media isregarded [23]. The following co-ordinate system is used:its origin is placed in the center of the inflow section. Theaxis Ox is directed along the domain towards the outflowsection. The axes Oy and Oz are placed, respectively,vertically and sideward. The lateral surfaces bounding thedomain are impermeable. Initially, almost all the domainis filled with the fluid to be displaced except for a tinyportion near the inflow cross-section. The displacing fluidflows via the cross-section at x ¼ 0; both fluids outflow viathe section at x ¼ L.

The pressure is assumed to be uniform both at theinflow and outflow cross-sections being not constant intime. We will study a constant rate problem, i.e. a fluidflux at the inflow section is set as a problem parameter.Both fluids are assumed incompressible; the porousskeleton is assumed immobile. The skeleton porosity isconstant as well as the permeability. The gravity and othermass forces are neglected.

1.2. Scaling formulae

We choose the scaling factors denoted with a subscripts, as follows:

Us ¼ U; Ps ¼ m1LU=K; ts ¼ Lf=U; xs ¼ ys ¼ zs ¼ L;

Vs ¼Ws ¼ Us;ms ¼ m1: ð1Þ

The following notations are used in (1): U is the meanvelocity of filtration via inflow cross-section; L the lengthof the domain; m1 the displacing fluid viscosity; K thepermeability; f the porosity.

1.3. Governing equations

After the scaling is applied, the following dimension-less equations are obtained [12]

@s

@tþr � ðsvÞ ¼ r � ðDðvÞ � rsÞ; ð2Þ

mðsÞv ¼ �rp; ð3Þ

r � v ¼ 0: ð4Þ

Eq. (2) states for the displacing fluid saturation dynamics,Eq. (3) is the generalized Darcy law, and Eq. (4) is aconsequence of incompressibility of fluids and immobilityof the porous skeleton. The following notations ofdimensionless parameters are used in the system ofgoverning Eqs. (2)–(4):

s
saturation of the displacing fluid, ranging from 0 to 1
v
vector of the mean volumetric velocity of filtration
D
dispersion tensor, depending on velocity
m
effective fluids viscosity, depending on saturation
p
pressure

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Substituting (3) into (4), we obtain an elliptic equationfor pressure:

r � ðm�1rpÞ ¼ 0: ð5Þ

We will use this equation in our calculations.The dependence of the dimensionless dispersion

tensor on the velocity, in case of isotropic porous mediaand proportionality between magnitudes of both quan-tities are written in the tensor form [12,14,24,25]

DðvÞ ¼A

jvjðPe�1

L vvþ Pe�1T ðjvj

2E� vvÞÞ; ð6Þ

where vv is a dyadic product of velocity vector by itself, Eis the unit tensor of the 2nd range, A ¼ H=L is the ‘‘first’’aspect ratio of the domain, PeL and PeT are the long-itudinal and transversal P�eclet numbers built using thewidth of the domain H and the corresponding dimen-sional dispersion factors DL and DT :

PeL ¼H

DL; PeT ¼

H

DT: ð7Þ

In case both P�eclet numbers are equal, the expression (6)simplifies

DðvÞ ¼ APe�1jvjE: ð8Þ

Dispersion is this case is merely the diffusion withcoefficient proportional to velocity modulus. The mainreason to take the law (8) instead of simpler diffusion isincrease of fluids mixing with velocity in the porousmatrix, thus the diffusion coefficient is assumed to beproportional to velocity of filtration. It makes the diffusiveterm in the transport equation (2) non-linear.

In order to model the effective viscosity, we willuse the derivation of Darcy formula from the generalequations for multi-phase flows [26,27], in particular,Todd–Longstaff’s dependence [28,29], which was provedto be an effective one describing 1D approximation ofeffective fluids flux and pressure distribution in theprocess of displacement from porous media. Scaled usingthe displacing fluid viscosity as the scaling factor (1) thisdependence gives the following expression [28,29]:

1

m ¼ ðsþ ð1� sÞMo�1Þ � ðsþ ð1� sÞM�1=4Þ�4o;o ¼ 2

3: ð9Þ

Here, M ¼ m2=m1 is the viscosity ratio, and in case M41the displacement should be irregular. The mixing rule (9)is a type of combination of two power mixing rules usinga fit parameter o; various power mixing rules werestudied previously [12]. The power mixing rule with aparameter—an arbitrary real degree k is as follows:

m ¼ðsþ ð1� sÞMkÞ1=k; ka0;

M1�s; k ¼ 0:

(

Of the set of power mixing rules, most notable are thedefinite cases of the degree parameter: arithmetic averaging(k ¼ 1), which often models resulting viscosity in a free-flowfluids mixing, harmonic averaging (k ¼ �1), which corre-sponds to case of immiscible fluids situated in separateparallel pores and affected by same pressure gradient,exponential averaging (k ¼ 0), which was studied by Homsy,

Riaz, Riuth, Meiburg, Tchelepi and other authors, which usedthe spectral methods numerical approach [9,16–22].

1.4. Initial and boundary conditions

The initial conditions have a peculiarity: in order toinvoke instability in theoretical investigations one shouldintroduce it initially; otherwise the simulation couldchoose regular but unstable displacing mode. So, a slightlyirregular initial distribution of saturation is given near theinflow surface [23,30–31]:

t ¼ 0 : s ¼0; xZx0;

sx � x; xox0:

(ð10Þ

Here, x is a random field which magnitudes are dis-tributed uniformly from 0 to 1, sx is the random fieldscaling factor and x0 should be as close to zero as possible.At x0-0 or sx-0 we get close to the physical problemstatement: the full domain is filled with fluid to bedisplaced off. Both factors x0 and sx could be nameddestabilizing factors.

The boundary conditions for the constant rate case arethe following:

x ¼ 0 : s ¼ 1; p ¼ p0ðtÞ;

ZS0

1

m@p

@xdSþ aA ¼ 0; ð11aÞ

x ¼ 1 :@s

@x¼ 0; p ¼ 0; ð11bÞ

y ¼7A=2 :@s

@y¼ 0;

@p

@y¼ 0; ð11cÞ

z ¼7a=2 :@s

@z¼ 0;

@p

@z¼ 0: ð11dÞ

Here, S0 denotes the inflow cross-section. The boundaryconditions for pressure at x ¼ 0 (11a) are extended by anew unknown variable p0 and a new integral equationderived from the requirement of constant flux of thedisplacing fluid, inflow velocity being normalized to unity.

The governing equations for the corresponding 2Dproblem are the same as (2)–(5), because of their vectorform. The dependencies (6)–(8) for the dispersion and theviscosity mixing rule (9) are also the same, together withthe initial condition (10). The only difference is in theboundary conditions (11a) and (11d). The last conditionfades away together with the third co-ordinate and thefirst is replaced with the following:

x ¼ 0 : s ¼ 1; p ¼ p0ðtÞ;

Z A=2

�A=2

1

m@p

@x

� �x¼0

dyþ A ¼ 0: ð12Þ

1.5. Governing parameters

The set of dimensionless governing parameters, whichcontrol the solution of 3D displacement problem in ourpresent investigation, looks as follows:

M
viscosity ratio
Pe ¼ PeL ¼ PeT
P�eclet number
A, a
aspect ratios (primary and secondary)
sx ; x0
destabilizing factors

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We will fix the destabilizing factors and the primaryaspect ratio A. The secondary aspect ratio a, the viscosityratio M and the P�eclet number will be varied. The 2Dproblem that will be also studied, can be formally denotedwith vanishing a. The following values of governingparameters will be used in the present investigation:

M ¼ 3� 1000; Pe ¼ 50� 10 000; a¼ 0� 0:5; ðvariable parametersÞ;

A ¼ 1; sx ¼ 0:1; x0 ¼ 0:02ðconstant parametersÞ: ð13Þ

1.6. Integral parameters of the flow

In order to describe the most important features of thedisplacement problem, the following integral parametersof this process are considered.

The cross-section mean of the displacing fluid satura-tion. It is defined as follows:

cðx; tÞ ¼

ZS

s dS: ð14Þ

Parameter (14) is to be modeled in the section 1.10 belowby means of a simple 1D non-stationary model with twofit parameters.

The outflow, i.e. total convective flux of the displacingfluid at the outflow cross-section:

Q ðtÞ ¼

RS1

svx dSRS1

vx dS¼

ZS1

svx dS; ð15Þ

where S1 denotes the outflow surface. The equality in theexpression (15) is due to equivalence of the mean crosssection velocity to unity in the constant rate problemstatement.

The breakthrough time is determined as

Tb ¼minft : Q ðtÞ40g: ð16Þ

One can see that the breakthrough time corresponds tothe moment when the first finger reaches the outflowcross-section. If the displacement is regular and there isno displacing fluid in the domain at t=0, the dimensionlessbreakthrough time is about unity. With initial conditions(10), however, the breakthrough time is less than unityeven in the regular case; it tends to unity when thedestabilizing factors fade.

1.7. Numerical method

The calculations were processed on a rectangularuniform spatial grid. Most 3D calculations used 100�100� Nz cells, where Nz ¼ ½100a�. 2D calculations usedthe grid of 100� 100 cells. The time step interval Dt wasvariable in order to maintain CFL criterion.

At the first instance, the initial conditions are devel-oped. After that, successful transitions from one time layerto another are processed. The following algorithm wasused. At the initial instance (t ¼ 0):

�
The array of saturation s is formed using the initialconditions (10).
�
The initial estimate of pressure is generated. � The sequence of a time step calculation is processed,
beginning from the viscosity calculation stage (seebelow). Finally, the arrays of s; p;v are known at t ¼ 0.
� Transition between time layers is processed using the
following sequence.
� Initially, arrays of spatial distribution of s, p and v are
known for time t.
� Time interval Dt is calculated, using CFL criterion [32]. � The saturation s is promoted to a new time layer t þ Dt
using Eq. (2) and boundary conditions (11). In case oflow P�eclet number, the calculation is split intohyperbolic and parabolic parts. The hyperbolic partresponsible for convection is calculated using anexplicit TVD method [33] of the second order utilizingthe van Leer’s flux limiter [34]. The parabolic partresponsible for the dispersion is calculated implicitlyusing the PBCG techniques. In case of high P�ecletnumber, the dispersion term is applied explicitly. Thedispersion term is calculated using expression (8)(non-anisotropy case).
� The mobility variable m�1 is calculated at the new time
layer, using the expression (9) for Todd–Longstaff’seffective viscosity model [28,29].
� The pressure p is calculated at the new time layer,
using Eq. (5) and boundary conditions (11). The PBCGtechnique is used ; the previous pressure field is takenas the initial estimate. This is the most time consumingpart of calculations, because the elliptic equation mustbe solved as precisely as possible to avoid non-physicaleffect of finite velocity divergence and consequentsources of saturation.
� The velocity vector v is calculated at t þ Dt using the
Darcy law (3) with central-difference derivatives.

After each computation cycle, saturation, pressure andvelocity are known on the next time layer; the dispersiontensor and effective viscosity are also known being simplealgebraic expressions depending on the main parameters.The code was developed in C+[35].

Together with 3D case, we have also processedcalculations of 2D model coherent to the 3D one. In 2Dmodel, the second aspect ratio is set formally to zero:a ¼ 0, all the other parameters have the same definition.The sequence of 2D model calculation is similar, except forthe third velocity component and for the nature of grid,which is 2D. Also, one of the boundary conditions (11a) isreplaced by (12).

1.8. Validation of the numerical scheme

Validation of numerical scheme was performed byqualitative comparing the numerical 2D results withexperiments performed in Hele–Shaw cells [12] and withresults of numerical calculations undertaken using differ-ent numerical schemes. The direct quantitative compar-ison of results obtained using random initial conditions(10) is not a fair choice to validate the results, becausethose conditions depend directly on the grid parametersin our simulations. Therefore, in order to validate the

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scheme, we used initial conditions independent of themesh size and of the random seed. However, the initialsaturation should be non-uniform in order to obtain theunstable displacement.

We used the following initial conditions instead of (10),in our verification study:

s ¼1� 10x; xo0:1

0 otherwise

� �cos2ð10yÞ cos2ð10zÞ: ð17Þ

One should notice that the conditions (10) introduceharmonic variations into the initial saturation, which areexpected to produce fingers.

The artificial diffusion used in TVD method in order tosuppress non-physical oscillations of solutions in theregions of sharp gradients of parameters, depends directlyon the average size of the mesh, and its coefficient andartificial P�eclet number could be estimated as follows[33]:

Pe�1a jvj ¼

12jvjDx: ð18Þ

One should notice that the artificial P�eclet number is notequivalent to an ordinary P�eclet as if it is some effectivevalue, because the dispersion with artificial P�eclet isapplied only in the regions of sharp saturation gradients,and not elsewhere. Normally, it influences only the slopeinclination, but in case of unstable displacement, itspresence could influence the fingers growth as well. Note,that when the flux is promoted to the 2nd order ofaccuracy, it does not allow jumps to dissolve above somelevel, unlike a linear diffusive flux. Thus, the effect ofartificial diffusion is limited and not equal to an effect ofdiffusion at some P�eclet number.

Two series of investigations were processed in order toinvestigate the influence of the mesh size. First series usedthe grid 80�80�40 cells, second 100�100�50. All theother parameters were the same. Calculations wereprocessed for high viscosity ratio, medium P�eclet numberand high secondary aspect ratio: M ¼ 1000, Pe ¼ 1000,a ¼ 0:5. The initial conditions were (17), other features ofthe problem statement, as above. Parameters correspondto a highly unstable displacement case. The effectiveartificial P�eclet was 160 and 200, respectively. Results in

Fig. 1. Unstable displacement front pattern shown by surface sðx; y; zÞ ¼ 0:25 in

coarse one 80x80x40. Time moment is t ¼ 0:129.

Fig. 1 show 3D patterns of displacement front obtained bytime t ¼ 0:129 for both mesh cases (due to differentcalculation time interval, difference in times exists and islower than 0.001). The 3D displacement front is illustratedby a surface sðx; y; zÞ ¼ 0:25; the left picture corresponds toa fine mesh, the right to a coarse one.

Results shown in Fig. 2 illustrate qualitative similarityof the patterns; some details are slightly different,however. Yet many small features (e.g. secondaryfingers) are nearly the same. Symmetry is due tosymmetry of the initial and boundary conditions. Themain difference is the advanced fingers length; it is adirect influence of artificial diffusion. But the overalldifference of pictures is small: the error is within 3–5%related to completely displaced state.

The distributions of cross section mean saturationcorresponding to cases shown in Fig. 2 is shown in Fig. 3.Left and right plots relate to coarse and fine meshes; timemoment is the same. The fingers are advanced more forthe case of the fine grid (lower artificial diffusion), but theoverall difference of pictures is small: the error is within3–5% related to completely displaced state.

The results shown in Fig. 1 testify the expectabledependence of the flow patterns on the mesh size due toartificial diffusion effect. One of the remedies is usingspectral methods instead of the TVD approach to thetransport equation (2), which were applied to 2D and 3Ddisplacement problems in [9,16–22]. However, spectralapproaches have the similar feature—the more harmonicsare taken, the more accurate are the results (like more finegrid means better accuracy in our approach), though thenature of the error is different in different approaches.

1.9. Modeling integral parameters with 1D equation

The integral parameters of the flow can be estimatedusing a simple 1D one-equation model. This modelsimulates the dynamics of cross-section averaged satura-tion s (14). Such modeling was processed in [11,12]; it wasalso proposed in [28].

If we average the dynamics of saturation Eq. (2) in eachcross-section determined by longitudinal coordinate x, we

space. Left plot corresponds to the fine mesh 100x100x50, right to the

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Fig. 2. Patterns of the displacing front for lower viscosity ratio (M ¼ 10). Three subsequent time moments; Pe ¼ 1000, a ¼ 0:5.


will obtain the following model for s:

@s

@tþ@

@xðu � sÞ þ

@

@xðu0 � s0 Þ ¼ APe�1

L

@

@xu@s

@x; ð19Þ

where u is the mean velocity in the cross-section and u0

, s0

are deviations of the relative parameters from the cross-section means. The third term in Eq. (19) is a subject ofspecial modeling. Earlier [11], we have modeled it usingconvective (Buckley–Leverette) and diffusive terms multi-plied by model coefficients. It was shown, however, thatthis modeling cannot fit simultaneously both the long-runshape of the cumulative output curve (15) and thebreakthrough time. Then, introducing modifications intothe convective term, modeling of the outflow for 2D casewas processed [11] with much better fit.

We will model the additional convective term in (19)only with a convective component depending on averagedsaturation, but using two fitting parameters. One can

notice also that the dimensionless averaged velocity u isunity due to the constant rate case under investigation.We assume the following ad hoc dependence of theadditional convective term u0s0 on the mode parameters:

u0s0 ¼ au � sð1� sÞmaxfMg � 1;0g

1þ smaxfMg � 1;0g

� �: ð20Þ

Dependence (20) assumes that the additional flux takesplace only when both fluids are present in the same crosssection, that it cannot be opposite to average velocity, thatits maximum is regulated by constants a and g, which areto be modeled using comparison with multidimensionalcalculations.

Substituting (20) into (19) for M41, and taking intoaccount that u ¼ 1, we will obtain

@s

@tþ@

@xð1� aÞs þ a sMg

1þ sðMg � 1Þ

� �¼ APe�1

L

@2s

@x2: ð21Þ

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Fig. 3. Patterns of the displacing front for medium viscosity ratio (M ¼ 100). Three subsequent time moments; Pe ¼ 1000, a ¼ 0:5.


The initial and boundary conditions for (21) areobtained from (10) and (11), and look as follows:

sð0; xÞ ¼stð1� x=x0Þ; xox0;

0 otherwise;

(ð22Þ

sðt;0Þ ¼ 1; @s=@xjx¼1 ¼ 0: ð23Þ

Note that there is no specific reason to introducedisturbance parameters into 1D modeling but they areused in (22) in order to fit the initial conditions ofmultidimensional modeling.

The outflow obtained by 1D modeling is determined bythe convective term at x ¼ 1 due to its definition (20), (21)is calculated as follows:

qðtÞ ¼ ð1� aÞs þ a sMg

1þ sðMg � 1Þ

� �x¼1

: ð24Þ

The breakthrough time obtained by 1D modeling isdetermined similar to (16)

tb ¼minft : qðtÞ40g: ð25Þ

In order to calculate the fit parameters aand g depending on the governing parameters of thedisplacement process, we will use the following proce-dure.

�
Fix the governing parameters. � Process multidimensional modeling. � Choose three time moments tk, and obtain three
profiles of cross section mean saturationckðxÞ ¼ cðtk; xÞ, k ¼ 1;2;3.
� Obtain the outflow temporal profile Q ðtÞ and the
breakthrough time Tb.
� For a given set of fitting parameters a, g process 1D
modeling.
� For the time moments tk, obtain 3 profiles of the
average saturation skðxÞ ¼ sðtk; xÞ.
� Obtain the modeled outflow profile qðtÞ and the
breakthrough time tb.
� Calculate the function to be minimized, which is
chosen as follows:

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Fig. 4. Patterns of the displacing front for high viscosity ratio (M ¼ 1000). Three subsequent time moments; Pe ¼ 1000, a ¼ 0:5.


C ¼1

3

X3

k¼1

½

Z 1

0ðskðxÞ � ckðxÞÞ

2 dx�1=2

þ1

T

Z T

0ðqðtÞ � Q ðtÞÞ2 dt

� �1=2

þ ½Tb � tb�: ð26Þ

Our goal is to find the parameters a and g whichminimize the function C. This function depends both onthe governing parameters of the multidimensional model(parameters of the process) and on the fitting parametersa, g. Therefore, the fitting parameters minimizing C willdepend on the governing parameters of the model.

2. Results of investigations

2.1. Displacement front patterns

Using the parameters given in (13) and varying M, Pe

and a we have obtained solutions for the 3D and 2Dmodels. Figs. 2–4 illustrate flow patterns for different

viscosity ratio M ¼ 10;100;1000 and constant P�ecletnumber Pe ¼ 1000 and aspect ratio a ¼ 0:5. Each figureshows three flow patterns for different times; the flowpatterns are illustrated by the shape of the displacingfront which is determined as the surface with s ¼ 0:25.The right-hand and left-hand sides of each figure showthis surface from different points of view.

Fig. 2 shows that the fingers number is rather high,their shape is cylindrical, and they grow but do not split atthose values of parameters. One could see also that therear front of displacement is also moving visibly.

The next figure shows the displacement front patternsfor intermediate viscosity ratio.

It is seen from Fig. 3 that the fingers are thinner thanthose shown in Fig. 2 for lower M, their number is higher,their growth is less regular: some of them overcome others,and most advanced fingers split into a palm of daughterfingers. The rear front motion cannot be observed from thedisplacement front patterns shown in Fig. 3.

The next figure shows the displacement front for a veryhigh viscosity ratio.

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It is seen from Fig. 4 that the fingers are rather thin;also, they differ in size: some of the fingers merge intothick ones, and the last overcome other fingers; they formchannels from inflow to outflow surfaces of the domain.The displacement is highly irregular there, compared tothose shown in Figs. 2 and 3.

The next picture (Fig. 5) shows the influence of the P�ecletnumber. We fix the medium viscosity ratio (M ¼ 100) andcompare patterns of the front for different Pe at a ¼ 0:5.Now, we will not show the development of the front butsome intermediate state of it.

One can see from Fig. 5 that the P�eclet number affectsmostly the thickness of the fingers, together with theirnumber. The irregularity of displacement is high for all thethree cases shown in Fig. 5, because the fingers split andmerge for all those cases. However, the thickness of thefingers is much less for Pe ¼ 10 000 than for Pe ¼ 1000.

High irregularity of the displacing front is seen on eachplot in Fig. 5; however, the fingers number naturallygrows with the growth of the cross section area. For higha, development and splitting of the most advancingfingers looks similar; for low a, the front is bounded by

M = 100, Pe =

M = 100, Pe = 20

M = 100, Pe = 1

Fig. 5. Influence of the P�eclet number for intermediate

the walls, and the fingers split in lower number ofdaughter structures. Therefore, the secondary aspect ratiois expected to influence the displacement features mostlywhen it is low.

Patterns of 3D displacing front for M ¼ f10;100;1000gand Pe ¼ f1000;2000;10 000g are shown in Figs. 2–5. Theregion of M ¼ f3;10;100;1000g and Pe ¼ f50;100;200;500g is featured with the flow regularization area, so itis worth to look at the fully developed displacing processfor all of those parameters. Fig. 6 shows the displacementfront patterns obtained for a ¼ 0:5.

One should notice that Fig. 6 contains not thedevelopment of the displacement process but 16 patternsof the displacement front obtained from different sets ofgoverning parameters just before the displacing fluidbreaks through the outflow boundary of the domain.

It is seen from Fig. 6 that at M ¼ 3 the flow is regularfor all the bandwidth of 50rPer500, at M ¼ 10 slightirregularity of the displacement front (like a banneroscillations) is seen at Pe ¼ 200, and at Pe ¼ 500 the flowis definitely irregular. At M ¼ 100 the flow at Pe ¼ 50 isalmost regular, at Pe ¼ 100 and higher it is irregular.

1000

00

0000

viscosity ratio M ¼ 100. The aspect ratio a ¼ 0:5.

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Pe M = 3 M = 10 M = 100 M = 100050

100

200

500

Fig. 6. Patterns of the displacing front for M ¼ 3;10;100;1000 (rising from left to right) and Pe ¼ 50;100;200;500 (rising downwards). 3D flow, thickness

aspect ratio a ¼ 0:5.


At M ¼ 1000 the flow is irregular even at Pe ¼ 50. It is alsoworth to seen that the fingers in irregular flow patternsdecrease in thickness and increase in number withincreasing both the viscosity ratio M and the P�ecletnumber Pe.

Fig. 7 summarizes the flow patterns for 2Ddisplacement simulations illustrating the saturationpatterns for the displacing fluid. The flow patterns ineach row are made for the same P�eclet number anddifferent viscosity ratio M ¼ 10;100;1000 (from left toright). The following P�eclet numbers were tested:Pe ¼ 50;100;200;500;1000;2000;10 000. Each plot wastaken at some characteristic time moment.

Fig. 7 shows the same tendencies of M and Pe influenceon the displacement process as it was shown for 3Dprocess in Figs. 2 and 4. However, two significantdifferences take place. First, the total number of fingersis very small compared with 3D displacement at highaspect ratio. Second, for high viscosity ratio and P�ecletnumber, separation of tips of the fingers takes placeinstead of their split. We cannot observe this separationon the displacement front patterns for 3D case because wemonitor the surface of the displacement front there fors ¼ 0:25 only, but the separated fingers have much lesssaturation, as one can see in Fig. 8. Also, we did notpresented the case with both highest P�eclet number andviscosity ratio for 3D case.

2.2. Results for the fitting parameters of 1D modeling

The fitting parameters of 1D modeling [12] a and gwere obtained minimizing the function C (26) for each set

of parameters a;M; Pe for which multidimensional calcu-lations were processed. The optimized values of a and gwere obtained for each variant of the random seed, andthen the average values and deviations of fit parameterswere obtained as follows:

a ¼ 1

n

Xn

k¼1

ak; g ¼ ðYn

k¼1

gkÞ1=n;da ¼max

kjak � aj;dg

¼maxkjgk � gj: ð27Þ

Here, n is the number of variants, da and dg are deviations.One could notice that, we used arithmetic averaging fora and geometry averaging for g due to their specificposition in formula (20) modeling the additional con-vective term.

In order to illustrate the fitting quality by means ofcurves behavior, we place in Fig. 8 two sets of illustra-tions: for 2D case and lower P�eclet number (higher row),and for 3D case, a ¼ 0:5 and higher P�eclet number (lowerrow). The black curves in Fig. 8 relate to random seedvariants of the calculated outflow, the red curve is theirmean, and the green curve is obtained by 1D modeling forparameters a and g by means of minimizing C. Thequality of fitting is illustrated by degree of closenessbetween the red and green curves on each plot.

One could see from Fig. 8 that the quality of fit is betterfor 3D case than for 2D, and for lower viscosity ratiosrather than for higher ones. This phenomenon shows thatour 1D modeling has its borders of applicability; expand-ing those borders may result in introducing an additionalfit parameter, or some other features which are beyondthe scope of the present study.

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Pe = 50

Pe = 100

Pe = 200

Pe = 500

Pe = 1000

Pe = 2000

Pe = 10000

Fig. 7. Saturation of the displacing fluid for 2D displacement process. Flow patterns for different P�eclet numbers and viscosity ratios.


Fig. 9 shows plots for different M for intermediate a ¼ 14

and Pe ¼ 2000. Each row shows fitting the meansaturation profiles (left) and the outflow (right).

It is seen from Fig. 9 that fitting of both the profiles andthe outflow is good enough only for the lower viscosityratio. For the higher viscosity ratio, the outflow is modeled

good enough, but the profiles are modeled much better atx � 1 than for the lower values of x. This could beexplained by the fact the outflow plays a significant role inthe function to be minimized, and the outflow is formedby the saturation profile near the outflow cross-section atx ¼ 1, not by the saturation in other locations. At the

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Fig. 8. Examples of 1D modeling (the calculated and modeled outflow). Good or satisfactory quality of phenomenological parameters fit for chosen sets of

governing parameters.


highest M, the outflow modeling results in a considerabledeviation for 0:2rtr0:8, but both the long-run behaviorof the outflow and the breakthrough time are modeledrather good. In any case, this result proves the assumptionof the presence of borders of 1D model applicability forhigh viscosity ratio.

Joint results for the values of 1D modeling fitparameters a and g together with their dependence uponall the available M; Pe; a are shown in Figs. 10 and 11.Fig. 10 shows curves of the dependence aðPeÞ for fixed M

(left-hand side), and the dependence aðMÞ for fixed Pe

(right-hand side). The upper row shows plots for a ¼ 0:5,medium row—for a ¼ 0:25, bottom row illustrates resultsfor 2D case (a ¼ 0).

It is seen from the dependence aðPeÞ that while thedisplacement is regular for lower values of the P�ecletnumber, the values of a lay on some curve slightlydecreasing with Pe. Curves corresponding to differentviscosity ratio M do not coincide, but for fixed Pe theparameter a slightly decreases with M (this is seen alsofrom the right-hand side plots). When the P�eclet numberincreases, the regularity of displacement is violated, and aincreases sharply. Then, it passes its maximum and startsdecreasing. The right-hand side plots are not so illustra-tive; they show only that a grows with M in case of theirregular displacement.

The correspondent curves for the ‘‘degree’’ phenom-enological parameter g are shown in Fig. 11.

The behavior of the second phenomenological para-meter shown in Fig. 11 seems like oscillating around somecurve rising with Pe (left-hand plots of the figure). Theright-hand plots show that the dependence of thisparameter on the viscosity ratio is still unclear but

roughly g is oscillating around some constant. Changebetween the regular and irregular displacement modesdoes not influence parameter g so much as a. Analyzingthe dependence of g on Pe for different M (Fig. 11), wenoticed that it is good to assume independence of g on M

and linear dependence on lnPe. The dependence on thesecondary aspect ratio was assumed as dependence via acomplex b ¼ 2=ðaþ 1=aÞ because of the symmetry ofphysical solution at a ¼ 1. The following interpolationwas derived by least squares method:

g ¼ ð0:0492þ 0:0348bÞ � lnPe� ð0:117þ 0:173bÞ;

b ¼ 2=ðaþ 1=aÞ: ð28Þ

The ‘‘degree’’ parameter g does not depend on theviscosity ratio, and Fig. 10 shows that the parameter adependence on Pe is fair, but its dependence on M is unfairfor any analysis. Formula (20) shows that a is joint togetherwith g in it, which enabled to find better approximation forthe product G ¼ ag as a function of M, Pe and a using thesame template as in Fig. 10. The interpolation formula wasobtained using the least squares method; it is morecomplex than (28) due to dependence on three parametersand change from regular to irregular mode.

GðPe;M; aÞ

¼ max

0:070� 0:032 ln M þ 0:004 ln2 M;

0:278 lnðln Pe� ð0:356þ 0:251 ln M

þ0:148bþ 0:056b ln MÞÞ þ ð�0:444

þ0:050 ln M � 0:047bþ 0:019b ln MÞ

8>>>><>>>>:

9>>>>=>>>>;: ð29Þ

Changing two alternatives of the maximum operator in(29) corresponds to change from regular to irregular modeof displacement.

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Fig. 9. Fitting the mean saturation profiles and the outflow for a ¼ 1=4, Pe ¼ 2000 and different values of viscosity ratio M.


A good approximation of the modeling coefficientscannot in general guarantee the quality of final resultsapproximation, i.e. curves of the outflow versus time.Fig. 12 shows the results for the outflow obtained forapproximated 1D modeling parameters (blue curves)compared with the actual 1D modeling parametersobtained minimizing the difference between 1D andmultidimensional modeling (green curves) and with theresults of multidimensional modeling. In fact, it is a part ofFig. 8 with the results of approximation shown upon it.

One can see from Fig. 12 that a quality of approxima-tion obtained for the viscosity ratio up to M ¼ 100 is good;both fitting and approximation are poorer for M ¼ 1000(Fig. 8 shows it clearly). This result shows some borders ofapplicability of our method. The developed phenomen-ological model for the mixing flux is not as good as itshould be, and the problem of representing viscousfingering by an effective model is still open. However, ingeneral, we suppose our formulae (28) and (29) modelingthe fit parameters to be satisfactory within a wide range of

governing parameters, which allows to use it for practicalapplications.

2.3. The effect of initial disturbance on displacement

instability

The instable displacement depends on the initialdisturbances introduced into the condition (10) by meansof the random seed. Changing the random seed, oneobtains different variants of the flow for the same valuesof governing parameters. However, those variants pre-serve qualitative features of the displacement process. Thequantitative features differ, but the deviation could beestimated processing several variants.

For each set of governing parameters, we haveprocessed 3 variants differing only in the random seed.This helped us to estimate the deviation, which isintroduced by stochastic instabilities into such integralparameters of the flow, as the outflow of the displacing

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Fig. 10. Plots of fit parameter a dependence on P�eclet number, viscosity ratio and secondary aspect ratio calculated by minimizing the function C.


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Fig. 11. Plots of fit parameter g dependence on P�eclet number, viscosity ratio and secondary aspect ratio calculated by minimizing the function C.


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Fig. 12. Examples of 1D modeling (the calculated, the modeled outflow and the outflow modeled with approximated parameters).


fluid Q ðtÞ, which is determined by means of the expression(15).

The deviation estimate is calculated as follows. Let QiðtÞ

be an i-th outflow function determined on a time interval½0; Ti�, i ¼ 1;2;3. Only the random seed variations, and notthe governing parameters introduce the difference in Qi.Then Q ðtÞ is determined on the time interval ½0; T� as anarithmetic mean of the three outflow values:

Q ðtÞ ¼1

3

X3

i¼1

QiðtÞ; 0rtrT; T ¼ mini

Ti: ð30Þ

The deviation is estimated as a square root of the L2

difference between Qi and their average on the timeinterval ½0; T�:

D ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

3

X3

i¼1

Z T

0ðQiðtÞ � Q ðtÞÞ2 dt

vuut : ð31Þ

The deviation of the outflow obtained from the dataprocessed now and the data processed in [36] is presented

in Table 1. It is shown in percents, i.e. multiplied by 100.The notation n/a means unavailability of data.

Table 1 results show that in a regular displacementcase the deviation between different outflow patterns isvery small, not more than fraction of a percent. If thedisplacement front is irregular, then the deviation rises upto 7.8% (in the worst case presented in Table 1). However,the worst case is obtained not for the highest P�ecletnumber but for an intermediate one. The highest devia-tion is obtained for 2D case, not for 3D. It can be explainedby lower number of stochastic viscous fingers. For highernumber of fingers in 3D case, they compensate effectsintroduced by each other, and the total outflow comes tobe more regular. The highest deviation obtained in 3Dcase is 4.5%.

Figs. 13 and 14 below illustrate plots of the outflowversus time on the basis of t 2 ½0;2�. Each plot is obtainedfor a definite set of ðM;Pe; aÞ; different dark curves showthe actual Q ðtÞ for each variant of random seed, themedium contrast curve (red) shows the average outflow,and the light curve (green) shows deviation. The resultsare placed in rows with 3 pictures, and the rightmost

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Table 1Average deviation between the outflow functions differing in the random seed only.

Deviation D (%) M=3 M=0 M=100 M=1000

2D calculations

Pe=50 0.0019 0.0020 0.2858 4.5825

Pe=100 0.0032 0.0046 2.3335 7.0892

Pe=200 0.0043 1.0286 4.4172 7.8046

Pe=500 0.0466 2.9420 5.6604 4.2479

Pe=1000 N/a 4.4104 3.6620 1.9855

Pe=2000 N/a 4.0135 3.7242 2.6108

Pe=10 000 N/a 3.3850 4.3090 3.2816

3D calculations, a ¼ 0:25

Pe=50 0.0002 0.0002 0.0430 3.4009

Pe=100 0.0003 0.0014 1.4274 3.5955

Pe=200 0.0006 0.2064 2.9261 1.5937

Pe=500 0.0212 1.1939 2.3984 3.3933

Pe=1000 N/a 0.8423 1.9116 2.0133

Pe=2000 N/a 0.7749 3.7112 1.9814

Pe=10 000 N/a 0.9381 1.8712 1.9126

3D calculations, a ¼ 0:5

Pe=50 0.0001 0.0005 0.0520 2.4317

Pe=100 0.0002 0.0045 1.7282 4.5011

Pe=200 0.0006 0.0896 1.8748 3.5132

Pe=500 0.0121 1.0236 1.8900 2.5344

Pe=1000 N/a 0.7094 1.1574 2.6774

Pe=2000 N/a 0.5452 1.4517 1.5274

Pe=10 000 N/a 0.5846 0.9260 N/a

Shown for all the parameters of the flow, including data obtained in [6]. Data is shown in percents.


picture in each row corresponds to a result for 2Dmodeling.

Fig. 13 shows dependence on M and a (for anintermediate P�eclet number Pe ¼ 200), and Fig. 14 showsthe dependence on Pe and a, for M ¼ 10.

It is seen from Fig. 13 that increasing the viscosity ratioaffects the outflow much stronger than changing the‘‘second’’ aspect ratio (or transferring to 2D). The devia-tion between curves for random seed variants is almostzero for regular displacement (here, Mr10). For lower M,the outflow grows almost linearly after the breakthrough(at t � 0:8) until it reaches unit. For higher M, the outflowgrows very fast after the breakthrough (that takes placeearlier), but then the growth rate quickly decreases. Thedependence on a shows much more irregular outflow for2D case than for the 3D one. The local deviation reacheseven 30% near the breakthrough time for 2D case atM ¼ 1000. The flow patterns show much more irregularelements in 3D displacement front than in 2D one. But thehigh number of fingers in 3D case compared to 2D bringsto compensation of contribution from them into theoutflow, and thus the integral profile looks much moreregular in 3D cases.

The deviation from an average curve is highest near thebreakthrough time. Then it usually decreases, so the time-averaged deviation presented in Table 1 is usually lessthan its peaks at Fig. 13. The local deviation in 3D casesdoes not exceed 10% (compared to 30% in one of 2D cases).

Fig. 14 shows the outflow patterns for fixed viscosityratio M ¼ 10 and varying P�eclet number (Pe ¼ 50;500;10 000).

It is seen from Fig. 14 that increasing the P�eclet numberdoes not affect the outflow so much as increasing the

viscosity ratio. A small decrease in the rate of outflowgrowth one can observe only, together with the decreaseof the breakthrough time. The dependence on a is weak,but the irregularity grows definitely with the decrease ofthe ‘‘second’’ aspect ratio. Maximum of the local deviationin this series of the outflow dynamic patterns is obtainedfor 2D case and maximal P�eclet number, it is located notnear the breakthrough time, and is about 10%. Maximaldeviation for 3D cases does not exceed 5%.

3. Conclusions

Multi-dimensional (2D and 3D) calculations of theforced convection of two-phase fluids in encumberedspace were performed. The blockage of space by differentobstacles was simulated by using the model of porousmedium. The displacement process of viscous fluid by aless viscous one was studied for a set of governingparameters (viscosity ratio, P�eclet number and thesecondary aspect ratio). The outflow of the displacingfluid versus time was obtained, together with profiles ofthe cross-section mean saturation of the displacing fluid.

Comparison of the integral parameters developmentshowed that for the displacement percentage, the resultsfor a ¼ 0:5 differ significantly from the results for a ¼ 0:1(both cases) and the 2D case. The results for 2D modelingare close to the results of 3D at a ¼ 0:1. However, theresults for cumulative outflow of the displacing fluid differinsignificantly for all the cases investigated. The break-through of the displacing fluid takes place earlier forhigher values of the ‘‘second’’ aspect ratio a; for 2D casethe breakthrough time is the longest one. Thus flow

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Fig. 13. Plots of the outflow versus time for lower P�eclet number, for various M and a. The right-hand picture in each row relates for 2D modeling.


instability and displacement front irregularity developmuch faster in a 3D case then in a 2D one.

The multi-dimensional irregular displacement wasmodeled by 1D equation for the cross-section mean ofthe displacing fluid saturation. This model has 2 fittingparameters: a and g, which still depend on the governingproblem parameters, which testifies that the developedphenomenological model represents one possible approx-imation for the mixing flux, probably not the best one.However, the fitting parameters were found minimizing a

functional depending on difference between profiles, 1Dmodeled and obtained from multi-dimensional calcula-tions. The functional forms approximating dependence offitting parameters on Peclet number, aspect and viscosityratios were developed, making it possible to use thedeveloped model for a rather wide range of governingparameters variation, which are suitable for practicalapplications.

It was obtained that 2D displacement integral para-meters (such as the outflow of the displacing fluid) have

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Fig. 14. Plots of the outflow versus time for fixed viscosity ratio M ¼ 10, for various Pe and a. The right-hand picture in each row relates for 2D modeling.


much higher deviation than those parameters obtained in3D modeling. This can be explained by much highernumber of fingers in 3D cases of displacement reachingthe outflow surface, and by the subsequent compensationof disturbances in the overall outflow.

Analysis of the fitting parameters of 1D modeling showthat the ‘‘degree’’ parameter g depends mainly on P�ecletnumber and aspect ratio, and not on the viscosity ratio.The dependence of a complex of those parameters: G ¼ agon Pe and M is much clearer than the correspondingdependence of a. This results in the development of rathersimple interpolation formulae for g and G, which predictboth dependence on P�eclet number, viscosity ratio andsecondary aspect ratio, and the conditions of changebetween regular and irregular modes of displacement.

The instable displacement depends on the initialdisturbances introduced into the numerical simulations.In a regular displacement case the deviation betweendifferent outflow patterns obtained for different initialdisturbances is very small, not more than fraction of apercent. If the displacement front is irregular, then thedeviation rises up to 7.8%. The highest deviation is

obtained for 2D case, not for 3D. It can be explainedby lower number of stochastic viscous fingers. For highernumber of fingers in 3D case, they compensate effectsintroduced by each other, and the total outflow comesto be more regular. The highest deviation obtained in3D case is 4.5%. The time-averaged deviation is usuallyless than its local peaks. The local deviation in 3D casesdoes not exceed 10% (compared up to 30% in one of 2Dcases).

Acknowledgments

The present investigation was supported in parts byRussian Foundation for Basic Research Grants 09-08-12131 and 09-08-00265.

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