Description

Analytical dual mesh method for two-phase ﬂow through highly
heterogeneous porous media
D. Khoozan
a
, B. Firoozabadi
a,⇑
, D. Rashtchian
b
, M.A. Ashjari
a
a
Department of Mechanical Engineering, Sharif University of Technology, Tehran, Iran
b
Department of Chemical and Petroleum Engineering, Sharif University of Technology, Tehran, Iran
a r t i c l e i n f o
Article history:
Received 25 February 2010
Received in revised form 11 January 2011
Accepted 24 January 2011
Available onli

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Analytical dual mesh method for two-phase ﬂow through highlyheterogeneous porous media
D. Khoozan
a
, B. Firoozabadi
a,
⇑
, D. Rashtchian
b
, M.A. Ashjari
a
a
Department of Mechanical Engineering, Sharif University of Technology, Tehran, Iran
b
Department of Chemical and Petroleum Engineering, Sharif University of Technology, Tehran, Iran
a r t i c l e i n f o
Article history:
Received 25 February 2010Received in revised form 11 January 2011Accepted 24 January 2011Available online 3 February 2011This manuscript was handled by P. Baveye,Editor-in-Chief
Keywords:
Dual meshAnalyticalVorticityTwo-phase ﬂowMulti-scale methods
s u m m a r y
Detailed geological models of a reservoir may contain many more cells that can be handled by reservoirsimulators due to computer hardware limitations. Upscaling is introduced as an effective way to over-come this problem. However, recovery predictions performed on a coarser upscaled mesh are inevitablyless accurate than those performed on the initial ﬁne mesh. Dual mesh method is an approach that usesboth coarse and ﬁne grid information during simulation. In the reconstruction step of this method, theequations should be solved numerically within each coarse block, which is a time consuming process.Recently, a new coarse-grid generation technique based on the vorticity preservation concept has beenapplied successfully in the upscaling ﬁeld. Relaying on this technique for coarse-grid generation, a novelmethod is introduced in this paper, which replaces the time-consuming reconstruction step in the dualmesh method with a fast analytical solution. This method is tested on challenging test cases regardingupscaling, in order to examine its accuracy and speed. The results show that the simulation time isdecreasednoticeablywithrespecttotheconventionalsimulationmethods.Itisalso2–4timesfasterthanthe srcinal dual mesh method with almost the same accuracy.
2011 Elsevier B.V. All rights reserved.
1. Introduction
Advances in reservoir characterization technologies, such asmodern seismic facilities, and well logs, can provide very detailedgeostatistical models of a reservoir. These models would contain10
11
–10
18
cellsiftheyresolvedthereservoiratthecoreorlogscale(Renard and de Marsily, 1997). By contrast, typical reservoir simu-latorscanhandleuptoonly10
5
–10
6
simulationcellsdependingonthe type of simulation and the available computer hardware. Evenconsidering the advances in computer technologies, there exists awide gap between ﬁne geological models and the size supportedby traditional simulators. In addition, a typical reservoir engineer-ingstudymaycontainnumeroussimulations for historymatching,investigation of different well conﬁgurations, and the assessmentof uncertainties using multiple geostatistical realizations. There-fore, the ﬁne grid geological model is required to be upscaled toa coarse simulation model.Upscalingtechniquesareintroducedtocoarsenthesegeologicalmodelstomanageablelevelsforusinginreservoirsimulators.Dur-ing the process of upscaling, small-scale properties such as perme-ability and porosity are averaged over coarse blocks and replacedby overall upscaled or homogenized properties.Generally, there are two important issues regarding the upscal-ing process. The ﬁrst one is the method of the coarse-grid genera-tion and the second one is the method of averaging the propertiesovercoarse-gridblocks. Thesetwoissueshave agreat effect ontheupscalingerrors, i.e.homogenizationandnumericalerrors. Replac-ing the ﬁne scale properties with equivalent averaged ones resultsin homogenization error while numerical errors are due to an in-creaseinthesizeofgridblocks.Upscalingmethodsaimatdecreas-ing these two errors by introducing proper grid generationtechniques and accurate averaging methods.Single-phase upscaling is the simplest, most widely used andbest understood form of upscaling in which the focus is placedon the calculation of equivalent absolute permeability. A reviewof these techniques can be found in Wen and Gómez-Hernández(1996), Renard and de Marsily (1997) and Durlofsky (2003). Formoderatedegreesofcoarsening,single-phaseupscalingtechniquesoften provide acceptable results. At higher degrees of coarsening,sometypeofrelativepermeabilityandcapillarypressureupscalingis also generally required, which is known as two-phase upscaling(Chang and Mohanty, 1997; Hui and Durlofsky, 2005). In this pa-per, only single-phase upscaling will be used.Gridding methods could be categorized into three maingroups. Permeability-based gridding was ﬁrst introduced byGarcia et al. (1992). In their method, by introducing elasticgrids, a coarse grid is generated such that the permeability
0022-1694/$ - see front matter
2011 Elsevier B.V. All rights reserved.doi:10.1016/j.jhydrol.2011.01.042
⇑
Corresponding author. Tel.: +98 21 66165684; fax: +98 21 66000021.
E-mail address:
ﬁroozabadi@sharif.edu (B. Firoozabadi). Journal of Hydrology 400 (2011) 195–205
Contents lists available at ScienceDirect
Journal of Hydrology
journal homepage: www.elsevier.com/locate/jhydrol
variance within each coarse block is minimized. Many othervariations of this method have been investigated by severalresearchers (Farmer, 2002). The disadvantage of these tech-niques is that they are based on static information rather thandynamic information. As a result, they cannot capture the con-nectivity in the ﬁne scale.Flow-basedgriddingprocedureshavebeendevelopedbyavari-ety of investigators. Within a structured Cartesian framework,Durlofsky et al. (1996, 1997) presented a non-uniform coarse-gridgeneration technique that selectively removes ﬁne-scale grid linesin a manner that retains important high ﬂow regions. Castellini(2001) extended this method to curvilinear framework in three-dimensional systems. Using ﬂow as a base for gridding allowsthe coarse gridto capturemanyimportant features of the ﬁne-gridmodel.In ﬂow-based grids, the variation of permeability in coarse-gridblocks is not investigated. This led to a third group of grid genera-tion technique that uses both permeability variation and velocityﬁelds. In techniques of He (2004), He and Durlofsky (2006) andWenandGómez-Hernández(1997,1998), theﬂowandpermeabil-ity variations are taken into account in two separate steps. Vortic-ity-based gridding of Mahani (2005), Mahani and Muggeridge(2005),Mahanietal.(2009)andAshjarietal.(2007,2008)incorpo-ratesbothﬂowandpermeabilityeffectsusingasinglequantity,i.e.vorticity. Mostaghimi and Mahani (2010) compared the vorticity-based gridding technique to permeability-based and ﬂow-basedgridding techniques in a number of 2D heterogeneous models viasimulation of two-phase ﬂow on the constructed grids. They con-cluded that although performance of ﬂow-based and vorticity-based gridding is comparable in many cases, vorticity-basedgridding has the beneﬁt of producing coarse-grid blocks with amore uniform permeability and ﬂuid-properties distribution. Inthis paper, we will use Ashjari et al. (2007, 2008) method forcoarse-grid generation.As mentioned, implementing proper gridding techniques andaccurate averaging methods can result into a low homogenizationerror. However, the numerical error will still be a problem. Thisproblem can effectively be managed using multi-scale simulationtechniques.The ﬁrst multi-scale simulation technique was developed byRamé and Killough (1991). In their method, the pressure equationissolvedbyaﬁniteelementmethodonthecoarsegridandthenbyusing the spline interpolation, the ﬁne grid information was calcu-lated from the coarse grid. Then the conservation equations forﬂuid motion were solved on the ﬁne grid.In the dual mesh method of Guérillot and Verdière (1995) andVerdière and Guérillot (1996), the pressure ﬁeld is ﬁrst computedon the coarse grid and the velocity ﬁeld is then estimated withineach coarse block by solving for the pressure with approximateboundary conditions. Then the saturation is calculated on the ﬁnegrid. They reported a speed-up factor ranging from ﬁve to seven.Audigane and Blunt (2004) extended the dual mesh method of Verdière and Guérillot (1996) to three dimensions and includedgravity and wells. In their method, the pressure equation is solvedon the coarse grid using the IMPES method with a ﬁnite differencescheme. The pressure ﬁeld is then reconstructed on the ﬁne gridusing ﬂux boundary conditions from the coarse grid simulation.Then,thesaturationﬁeldisupdatedontheﬁnegridusingstandardsingle-point upstream weighting. They reported a speed-up factorof about four for their most complex 3D case. In the present work,the dual mesh method of Audigane and Blunt (2004) will be usedas a base.Firoozabadi et al. (2009) proposed an upscaling technique thataimed at reducing both homogenization and numerical errors bycombining the dual mesh method with vorticity-based gridding.In their method, the dual mesh method is applied to reducenumerical errors while vorticity-based gridding is used to dealwith homogenization error.Although Firoozabadi et al. (2009) applied the dual mesh simu-lation in vorticity-based grids; they did not use the main advanta-ges of vorticity-based generated grids. These advantages (whichwill be discussed later) convince incorporation of an analyticalsolution in the dual mesh method. Hence, a novel technique isachieved in this research that decreases the simulation timenoticeably while keeping the accuracy at the same time. The out-line of the paper is as follows: First, we brieﬂy describe the vortic-ity-based gridding technique and then the dual mesh method.Subsequently, the properties of the vorticity-based grids will bediscussedandthe analytical dual meshmethodwill be introduced.Finally, the performance of the proposed method will be evaluatedthrough three test cases.
2. Vorticity-based gridding
Vorticity,
~
x
,isavectordescribingtherateanddirectionofrota-tion of a ﬂuid particle at any point and mathematically is deﬁnedas the curl of velocity ﬁeld
ð
V
!
Þ
. Vorticity is a measure of how fasta particle changes its velocity direction while it travels in the ﬂowﬁeld(Mahani, 2005). For instance, in two-dimensional ﬂows in
x
–
y
plane,thevorticityvectorcanbeexpressedonlybyonecomponentin the
z
-direction:
~
x
¼
r
V
!
¼
@
v
y
@
x
@
v
x
@
y
~
k
ð
1
Þ
where
v
x
and
v
y
are the velocity components in
x
and
y
directionsrespectively and
~
k
is the unit vector in the
z
-direction. In single-phase ﬂow in isotropic homogeneous porous media, using Darcy’sLaw, vorticity can be expressed as (Ashjari et al., 2008):
~
x
¼
V
!
r
ln
K
ð
2
Þ
This equation shows that the vorticity vector is a function of totalvelocity and the gradient of logarithm of permeability. Here, theﬁne-scale permeability is assumed a diagonal and isotropic tensorthat can be considered a scalar.
According to Eq. (1), vorticity will be at its maximumin regionswith high ﬂow rates perpendicular to high permeability gradients.As an example, vorticity is negligible for ﬂow in homogeneousmedia or for ﬂow perpendicular to bedding and is signiﬁcantaround boundaries of layers for ﬂow parallel to layering in strati-ﬁed formations or for ﬂow in channelized systems.It is worth mentioning that in channelized systems, vorticity islarge along the channel boundaries where there is a high perme-abilityvariationperpendicular tothe velocityinthechannel; how-ever, inside the channel, vorticity can be very small because of alow permeability variation. Hence, it can be concluded that vortic-ity is capable of capturing connectivity and identifying the bound-aries of connected regions.Inadditiontotheabove,weclearlyseethatvorticityintensityisa good measure of (key) heterogeneities due to its dependence onthe gradient of logarithmof permeability. By inspecting the vortic-ity map, we can distinguish between the most important perme-ability layer contrasts and less important ones. We can alsorecognize areas with low permeability variation (which can becoarsened) as well as very heterogeneous areas (where we keepthegridsreﬁned).Thisisquiteimportantforsuccessfulcoarse-gridgeneration and upscaling. According to Ashjari et al. (2007, 2008),the algorithm of vorticity-based gridding is as follows:1. A ﬁne-scale single-phase ﬂow simulation is performed usingthe geological model. From this solution, the velocity ﬁeld isextracted for use in the next step.
196
D. Khoozan et al./Journal of Hydrology 400 (2011) 195–205
2. The vorticity map is generated from the obtained velocity ﬁeld.For this purpose, Eq. (1) has to be discretized using a ﬁnite dif-ference scheme. In addition, a simple linear velocity interpola-tion for estimation of tangential velocities at each grid blockboundary is required. The calculated vorticities by this methodwill have different values depending upon the permeabilitycontrast between layers, allowing the algorithm to distinguishbetween the most important layer permeability contrasts andless important ones.3. From vorticity distribution, coarse-grid structure is optimizedwhereby boundaries of grids are adapted based on recognizingareas of high and low vorticity variation. In this step, vorticitycutoffs(onecutoffforeachﬂowdirection)areselectedtocontroltheupscalinglevelandtodecidewhereﬁne-gridcellsshouldbemerged or retained. Grid blocks whose vorticity variation issmallerthanthecutoffaremergedtogethertomakecoarsergridblocks, and if their value is larger than the cutoff, they are keptreﬁned. Deﬁnitely, the generated coarse grid in this manner isstrongly dependent on the distribution of ﬁne grid vorticity,which in turn, depends on the calculated velocity ﬁeld.4. The permeability ﬁeld is upscaled for the generated coarse grid.In this work, permeability is upscaled using geometric averag-ing whereby a diagonal isotropic upscaled permeability isobtained.The resulting coarse-grid model has a non-uniformdistributionwithcoarsergridblocksinareasoflowvorticityvariationandﬁnergrid blocks in areas of high vorticity variation. However, the algo-rithm does not give unique coarse-grid distribution for a givenﬁne-grid model. Usually, there is an optimum coarse grid, whichcan best preserve vorticity. This can be identiﬁed by incorporatingvorticity map preservation error (
Evmp
). This is the error betweentwo vorticity maps, one obtained fromthe ﬁne-grid model and theother from the coarse-grid model. That is:
E
v
mp
ð
%
Þ¼
P
N
1
x
f z
x
r z
P
N
1
x
f z
100
ð
3
Þ
where
x
f z
and
x
r z
arethevorticitydistributionoftheﬁneandthere-ﬁned (a model which has the same resolution as the ﬁne grid whileits permeability distribution is replaced fromthe coarse grid) grids,respectively and
N
is the number of ﬁne-grid blocks.
Thisdeﬁnitionof
Evmp
isslightlydifferentfromthedeﬁnitiongi-venbyAshjarietal.(2007,2008).Itusestheadvantagesofbothrel-ative and absolute error deﬁnitions while removing the pitfalls of each. For example, in the near zero vorticity regions (which maynot be an important zone), the relative error will be large which isnot desired. In addition, the main disadvantage of absolute error isits inability to present the importance of the error with respect tothe base model. However, the proposed deﬁnition does not sufferfromthese two important problems. It is a relative absolute error.
Evmp
indicatestheextentbywhichthereferencevorticitymapispreservedbythecoarse-gridmodel.Basedonthevorticitymappres-ervation concept (Ashjari et al., 2007, 2008), low
Evmp
means lowinformation loss through coarsening and hence better upscaling.Since
x
r z
is computed on the reﬁned grid, it represents mainlythe homogenization error of the upscaled model caused by replac-ing the ﬁne-scale permeability with the equivalent coarse-scalepermeability. Hence, the obtained
Evmp
is called reﬁned-basedvorticity map preservation error (Firoozabadi et al., 2009).
3. Dual mesh method
After obtaining the coarse-grid model with a minimumhomog-enizationerror, we need to solve the ﬂowequations. To reducethenumerical dispersion errors, multi-scale methods are introduced.In this work, the dual mesh method of Audigane and Blunt(2004) will be used. This method has the following steps:1. The average saturation for each coarse grid is calculated.2. For the current time step, the coarse-scale pressure equation issolved numerically using the ﬁnite difference scheme whichyields coarse-grid ﬂuxes (
Q
) normal to the coarse-gridboundaries.3. Fine-grid ﬂuxes (
q
) on the boundaries of each coarse-grid blockare obtained approximately using the calculated coarse ﬂuxes.This is achieved by a kind of downscaling step, assuming trans-missibility weighting along the coarse boundary.4. Alocaltwo-phaseﬁne-scalepressureequationissolvednumer-ically over each coarse-grid block at each time step with theapproximate ﬁne-grid ﬂuxes as Neumann boundary conditionsand the coarse block pressureas a Dirichlet boundaryconditionfor obtaining a unique solution. Incorporating Darcy’s Law, thisreconstructs the ﬁne-scale ﬂuxes throughout the model. Theﬁne-grid ﬂuxes reconstructed in this manner conserve massbalance over the local (ﬁne-grid cells within a coarse block)and global (entire reservoir) domains.5. Finally, the saturation ﬁeld is updated from the reconstructedﬁne-scale velocity ﬁeld.The above algorithm, as shown by Audigane and Blunt (2004),signiﬁcantly reduces numerical errors of the coarse-scale simula-tion. However, it is unable to remove the homogenization errorresulting from assigning improper upscaled permeabilities tocoarse-grid blocks.Firoozabadietal.(2009)solvedthisproblembyintroducingvor-ticity-based generated grids into the dual mesh method to reducehomogenization and numerical dispersion errors simultaneously.However, they did not use the advantages of the vorticity-basedgridding techniques to improve the dual mesh method. In the nextsection, these advantages will be investigated and usedto enhancethe simulation speed of the dual mesh method.
4. Analytical dual mesh method
Inthestep4ofthedual meshmethod(asdescribedearlier), thepressure equation is solved numerically over each coarse-gridblock. This step is the most time consuming element of the meth-od. The basic idea of this paper is to replace the time consumingnumerical solution with a fast analytical one. However, no analyt-ical solution is available for a general case due to heterogeneity.Here, we will introduce some reasonable assumptions, which willlead to an analytical solution.
4.1. Assumptions
In the dual mesh method of Audigane and Blunt (2004), bothﬁne and coarse grid information are used. Hence, this methodnotonlyreducesthenumericalerrors,butalsoreducesthehomog-enization error. However, what should be done if it is desired to just reduce the numerical errors and treat the homogenization er-ror by upscaling methods?The basic answer to this question would be using the reﬁnedgrid instead of the coarse grid. As mentioned before, the reﬁnedgrid has the same resolution as the ﬁne grid while its permeabilitydistribution is replaced from the coarse grid. Since the grid size of the reﬁned grid is the same as the ﬁne grid, the numerical errorswill be minimum. However, the problem here would be the simu-lation speed. The simulation time over the reﬁned and ﬁne gridswould be approximately in the same order of magnitude.
D. Khoozan et al./Journal of Hydrology 400 (2011) 195–205
197
To enhance the simulation speed, one may apply the dual meshmethod over the reﬁned grid. While increasing the simulationspeed, the numerical errors are also reduced in this approach.Ourmethodisalsoadualmeshmethodthatisbasedonthereﬁnedgrid.To develop our method, ﬁrst we will investigate the basic fea-tures of the reﬁned grid. In this grid, the permeability ﬁeld of eachcoarse-grid block (over which the pressure equation is solved instep 4 of the dual mesh method) is homogeneous since they allhave the value of the block permeability at the coarse scale.Wealsousevorticity-basedgriddingfor coarse-gridgeneration.Because of its high accuracy (Ashjari et al., 2008), we leave thehomogenization error to be treated by this method. As mentioned,vorticity depends on both permeability and velocity ﬁelds. Hence,wecanassumethatthevelocityﬁeldovereachcoarse-gridblockisapproximately homogeneous. Since saturation is calculated di-rectly from the velocity ﬁeld, we can assume that the saturationﬁeld over each coarse-grid block is also homogeneous. This wouldbe our only approximation, which its accuracy will be shown to behigh by examining the method over simple and complicated testcases.Inthispaper,geometricaveragingisusedforcalculatingtheup-scaled permeability ﬁeld. Since geometric averaging results inequal permeability magnitude in
x
and
y
directions, the resultedpermeability ﬁeld will be isotropic.We also restrict ourselves to horizontal two-dimensional testcases and hence gravity effects are neglected. The capillary pres-sure effect is also neglected. Fluids and rock are also assumedincompressible which is a reasonable approximation for mostcases. Therefore, we can state the basis of our method as follows:
Simulation is performed over the reﬁned grid.
Saturationﬁeld is assumed homogeneous over each coarse-gridblock.
Upscaled permeability ﬁeld is isotropic.
The effects of gravity (horizontal two-dimensional cases) andcapillary pressure are neglected.
Rock and ﬂuids are assumed incompressible.Using these assumptions, we will derive the governing equa-tions of two-phase ﬂow in coarse-grid blocks in the next section.
4.2. Governing equations
Fig. 1 shows a coarse-grid block (with index
I
,
J
in the coarsegrid) with its boundary conditions in which the pressure equationshould be solved in the reconstruction step (step 4) of the dualmesh method. The length of the block in
x
and
y
directions are as-sumed
a
and
b
, respectively.The governing equations for two-phase ﬂow in porous mediaare:
r
:
Kk
ri
l
i
ð
r
P
i
q
i
g
r
D
Þ
¼
@
ð
/
S
i
Þ
@
t i
¼
n
;
w
ð
4
Þ
S
n
þ
S
w
¼
1
ð
5
Þ
Neglecting the capillary pressure and gravity effects, Eq. (4) yields:
r
:
Kk
ri
l
i
r
P
¼
@
ð
/
S
i
Þ
@
t
;
i
¼
n
;
w
ð
6
Þ
Here, we deal with the reﬁned grid, so permeability is homoge-neous over the block. It is assumed that the permeability ﬁeld atthe coarse scale is isotropic, hence:
K
¼
cte
¼
K
I
;
J
ð
7
Þ
where
K
I
,
J
is the absolute permeability of the block at the coarsescale and
cte
means constant. It is also assumed that the saturationﬁeld is homogeneous over the coarse blocks. Since relative perme-ability,
k
r
, is a direct function of the phase saturation, one mayobtain:
k
rn
¼
k
rn
S
c nI
;
J
¼
cte k
rw
¼
k
rw
S
c wI
;
J
¼
cte
ð
8
Þ
where
S
c nI
;
J
and
S
c wI
;
J
are the non-wetting phase and wetting phasesaturationsoftheblockatthecoarsescale,respectively.Theincom-pressibility of rock and ﬂuids yields:
l
¼
cte
q
¼
cte
/
¼
cte
ð
9
Þ
Introducing Eqs. (7)–(9) into Eq. (6) results in:
Kk
ri
l
i
r
2
P
¼
/
@
S
i
@
t i
¼
n
;
w
ð
10
Þ
As described before, in the reconstruction step of the dual meshmethod, the pressure equation is solved over the coarse blocks.Therefore, we should derive the pressure equation too. To do so,we sum Eq. (10) over the wetting and non-wetting phases. That is,
Kk
rn
l
n
þ
Kk
rw
l
w
r
2
P
¼
/
@
ð
S
n
þ
S
w
Þ
@
t
ð
11
Þ
However, since we have
S
n
+
S
w
=1, one may obtain:
@
ð
S
n
þ
S
w
Þ
@
t
¼
0
ð
12
Þ
Introducing Eq. (12) into Eq. (11), the ﬁnal form of the pressure
equation will be obtained as:
r
2
P
¼
0
ð
13
Þ
which is the well-known Laplace equation. The analytical solutionto this equation is available for different boundary conditions. Inthe next step, the boundary conditions will be derived.
4.3. Boundary conditions
Let us consider the block shown in Fig. 1. It is assumed that theblock has a thickness of
h
. Taking
Q
as the total ﬂow rate, we canwrite (using Darcy’s law):
Q
i
¼
Q
in
þ
Q
iw
¼
k
rn
l
n
þ
k
rw
l
w
Kbh
@
P
@
x
i
;
i
¼
1
;
2
Q
i
¼
Q
in
þ
Q
iw
¼
k
rn
l
n
þ
k
rw
l
w
Kah
@
P
@
y
i
;
i
¼
3
;
4
ð
14
Þ
where subscripts 1, 2, 3 and 4 refers to the left, right, top and bot-tomboundaries, respectively. RearrangingEq. (14), onemayobtain:
f
i
¼
@
P
@
x
i
¼
Q
ik
rn
l
n
þ
k
rw
l
w
Kbh
;
i
¼
1
;
2
f
i
¼
@
P
@
y
i
¼
Q
ik
rn
l
n
þ
k
rw
l
w
Kah
;
i
¼
3
;
4
ð
15
Þ
where
f
1
–
f
4
are the pressure gradients across the block boundaries.Since the problem should have a unique solution, the pressure at
Fig. 1.
A typical coarse-grid block.198
D. Khoozan et al./Journal of Hydrology 400 (2011) 195–205

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