Journal of Hydrology 400 (2011) 195–205

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Journal of Hydrology

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

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, Iranb Department of Chemical and Petroleum Engineering, Sharif University of Technology, Tehran, Iran

a r t i c l e i n f o s u m m a r y

Article history:Received 25 February 2010Received in revised form 11 January 2011Accepted 24 January 2011Available online 3 February 2011

This manuscript was handled by P. Baveye,Editor-in-Chief

Keywords:Dual meshAnalyticalVorticityTwo-phase flowMulti-scale methods

0022-1694/$ - see front matter � 2011 Elsevier B.V. Adoi:10.1016/j.jhydrol.2011.01.042

⇑ Corresponding author. Tel.: +98 21 66165684; faxE-mail address: firoozabadi@sharif.edu (B. Firooza

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 isdecreased noticeably with respect to the conventional simulation methods. It is also 2–4 times faster thanthe original 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 contain1011–1018 cells if they resolved the reservoir at the core or log scale(Renard and de Marsily, 1997). By contrast, typical reservoir simu-lators can handle up to only 105–106 simulation cells depending onthe 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-ing study may contain numerous simulations for history matching,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.

Upscaling techniques are introduced to coarsen these geologicalmodels to manageable levels for using in reservoir simulators. 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.

ll rights reserved.

: +98 21 66000021.badi).

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 propertiesover coarse-grid blocks. These two issues have a great effect on theupscaling errors, i.e. homogenization and numerical errors. Replac-ing the fine scale properties with equivalent averaged ones resultsin homogenization error while numerical errors are due to an in-crease in the size of grid blocks. Upscaling methods aim at decreas-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). Formoderate degrees of coarsening, single-phase upscaling techniquesoften provide acceptable results. At higher degrees of coarsening,some type of relative permeability and capillary pressure upscalingis 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

196 D. Khoozan et al. / Journal of Hydrology 400 (2011) 195–205

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-based gridding procedures have been developed by a vari-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 grid to capture many important 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) andWen and Gómez-Hernández (1997, 1998), the flow and permeabil-ity variations are taken into account in two separate steps. Vortic-ity-based gridding of Mahani (2005), Mahani and Muggeridge(2005), Mahani et al. (2009) and Ashjari et al. (2007, 2008) incorpo-rates both flow and permeability effects using a single quantity, 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 equationis solved by a finite element method on the coarse grid and then byusing 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 ofVerdière 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, the saturation field is updated on the fine grid using standardsingle-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 reduce

numerical 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 bediscussed and the analytical dual mesh method will be introduced.Finally, the performance of the proposed method will be evaluatedthrough three test cases.

2. Vorticity-based gridding

Vorticity, ~x, is a vector describing the rate and direction of rota-tion of a fluid particle at any point and mathematically is definedas the curl of velocity field ðV

!Þ. Vorticity is a measure of how fast

a particle changes its velocity direction while it travels in the flowfield (Mahani, 2005). For instance, in two-dimensional flows in x–yplane, the vorticity vector can be expressed only by one componentin the z-direction:

~x ¼ r� V!¼ @vy

@x� @vx

@y

� �~k ð1Þ

where vx and vy 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 ð2Þ

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 maximum in 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-ability variation perpendicular to the velocity in the channel; 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.

In addition to the above, we clearly see that vorticity intensity isa good measure of (key) heterogeneities due to its dependence onthe gradient of logarithm of 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 keepthe grids refined). This is quite important for successful coarse-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.

D. Khoozan et al. / Journal of Hydrology 400 (2011) 195–205 197

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 (one cutoff for each flow direction) are selected to controlthe upscaling level and to decide where fine-grid cells should bemerged or retained. Grid blocks whose vorticity variation issmaller than the cutoff are merged together to make coarser gridblocks, 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-uniform distributionwith coarser grid blocks in areas of low vorticity variation and finergrid 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 from the fine-grid model and theother from the coarse-grid model. That is:

Evmpð%Þ ¼PN

1 xfz �xr

z

��� ���PN

1 xfz

��� ��� � 100 ð3Þ

where xfz and xr

z are the vorticity distribution of the fine and the re-fined (a model which has the same resolution as the fine grid whileits permeability distribution is replaced from the coarse grid) grids,respectively and N is the number of fine-grid blocks.

This definition of Evmp is slightly different from the definition gi-ven by Ashjari et al. (2007, 2008). It uses the advantages of both rel-ative and absolute error definitions while removing the pitfalls ofeach. 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 sufferfrom these two important problems. It is a relative absolute error.

Evmp indicates the extent by which the reference vorticity map ispreserved by the coarse-grid model. Based on the vorticity map pres-ervation concept (Ashjari et al., 2007, 2008), low Evmp means lowinformation loss through coarsening and hence better upscaling.

Since xrz is computed on the refined grid, it represents mainly

the 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 minimum homog-enization error, we need to solve the flow equations. To reduce the

numerical 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 is

solved 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. A local two-phase fine-scale pressure equation is solved numer-ically over each coarse-grid block at each time step with theapproximate fine-grid fluxes as Neumann boundary conditionsand the coarse block pressure as a Dirichlet boundary conditionfor 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.

Firoozabadi et al. (2009) solved this problem by introducing vor-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 used to enhancethe simulation speed of the dual mesh method.

4. Analytical dual mesh method

In the step 4 of the dual mesh method (as described earlier), 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 methodnot only reduces the numerical errors, but also reduces the homog-enization error. However, what should be done if it is desired tojust 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 ofthe 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.

198 D. Khoozan et al. / Journal of Hydrology 400 (2011) 195–205

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.Our method is also a dual mesh method that is based on the refinedgrid.

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.

We also use vorticity-based gridding for coarse-grid generation.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,we can assume that the velocity field over each coarse-grid block isapproximately 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.

In this paper, geometric averaging is used for calculating the up-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.� Saturation field is assumed homogeneous over each coarse-grid

block.� Upscaled permeability field is isotropic.� The effects of gravity (horizontal two-dimensional cases) and

capillary 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: Kkri

liðrPi � qigrDÞ

� �¼ @ð/SiÞ

@ti ¼ n;w ð4Þ

Sn þ Sw ¼ 1 ð5Þ

Fig. 1. A typical coarse-grid block.

Neglecting the capillary pressure and gravity effects, Eq. (4) yields:

r: Kkri

lirP

� �¼ @ð/SiÞ

@t; i ¼ n;w ð6Þ

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 KI;J ð7Þ

where KI,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, kr, is a direct function of the phase saturation, one mayobtain:

krn ¼ krn ScnI;J

� �¼ cte krw ¼ krw Sc

wI;J

� �¼ cte ð8Þ

where ScnI;J and Sc

wI;J are the non-wetting phase and wetting phasesaturations of the block at the coarse scale, respectively. The incom-pressibility of rock and fluids yields:

l ¼ cte q ¼ cte / ¼ cte ð9Þ

Introducing Eqs. (7)–(9) into Eq. (6) results in:

Kkri

lir2P ¼ /

@Si

@ti ¼ n;w ð10Þ

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,

Kkrn

lnþ Kkrw

lw

� �r2P ¼ /

@ðSn þ SwÞ@t

ð11Þ

However, since we have Sn + Sw = 1, one may obtain:

@ðSn þ SwÞ@t

¼ 0 ð12Þ

Introducing Eq. (12) into Eq. (11), the final form of the pressureequation will be obtained as:

r2P ¼ 0 ð13Þ

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 Qin þ Q iw ¼ �krn

lnþ krw

lw

� �Kbh

@P@x

����i

; i ¼ 1;2

Q i ¼ Qin þ Q iw ¼ �krn

lnþ krw

lw

� �Kah

@P@y

����i

; i ¼ 3;4ð14Þ

where subscripts 1, 2, 3 and 4 refers to the left, right, top and bot-tom boundaries, respectively. Rearranging Eq. (14), one may obtain:

fi ¼@P@x

����i

¼ � Q i

krnlnþ krw

lw

� �Kbh

; i ¼ 1;2

fi ¼@P@y

����i

¼ � Qi

krnlnþ krw

lw

� �Kah

; i ¼ 3;4ð15Þ

where f1–f4 are the pressure gradients across the block boundaries.Since the problem should have a unique solution, the pressure at

D. Khoozan et al. / Journal of Hydrology 400 (2011) 195–205 199

one point in the domain should be specified. Therefore, we shouldapply the last boundary condition. That is, the pressure at the centerof the block should be equal to the pressure of the coarse block re-sulted from the coarse grid solution. That is,

Pa2;b2

� �¼ Pc

I;J ð16Þ

where PcI;J is the pressure of the block at the coarse scale solution. By

defining the boundary conditions, the problem is closed and we canderive the solution.

4.4. Analytical solution

Consider the Laplace equation in two dimensions. The solutiondomain and the Neumann boundary conditions are shown in Fig. 2.The necessary and sufficient condition for solvability of the prob-lem has the form (Polyanin, 2002):Z b

0f1ðyÞdy�

Z b

0f2ðyÞdyþ

Z a

0f3ðxÞdx�

Z a

0f4ðxÞdx ¼ 0 ð17Þ

To solve our problem, first we should check the solvability con-dition, i.e. Eq. (17). Introducing the parameters f1–f4 derived in theprevious section into Eq. (17), one may obtain:

1krnlnþ krw

lw

� �Khð�Q 1 þ Q 2 � Q 3 þ Q 4Þ ¼ 0 ð18Þ

Hence:

Q 1 � Q 2 þ Q 3 � Q 4 ¼ 0 ð19Þ

As it is clear, the above equation is the mass conservation equationover the coarse-grid block. The fluxes derived from the coarse scalesolution will definitely satisfy this equation since the material bal-ance is satisfied when the pressure equation is solved over thecoarse grid.

Next, we should derive the solution. The general solution of thisproblem is presented in Eq. (20) (Polyanin, 2002). The definition ofAn–Dn can be found in Polyanin (2002).

Pðx; yÞ ¼ � A0

4aðx� aÞ2 þ B0

4ax2 � C0

4bðy� bÞ2 � D0

4by2 þ K � b

�X1n¼1

An

kncosh

npbða� xÞ

h icos

npb

y� �

� bX1n¼1

Bn

kn

� coshnpb

x� �

cosnpb

y� �

� aX1n¼1

Cn

kn

� cosnpa

x� �

coshnpaðb� yÞ

h i� a

X1n¼1

Dn

kn

� cosnpa

x� �

coshnpa

y� �

ð20Þ

After calculation and simplification of the coefficients, the solutioncan be obtained as:

Pðx; yÞ ¼ � f1

2aðx� aÞ2 þ f2

2ax2 � f3

2bðy� bÞ2 þ f4

2by2 þ const ð21Þ

Fig. 2. The domain of the problem.

The solution is surprisingly simple! The only remained coefficient isconst. To calculate this coefficient, we recall the Dirichlet boundarycondition, Eq. (16). Introducing this condition to Eq. (21), one mayobtain:

const ¼ PcI;J þ

a8

f1 � f2ð Þ þ b8

f3 � f4ð Þ ð22Þ

which completes the solution. Now we will implement this solutionto the dual mesh method to obtain the analytical dual meshmethod.

4.5. Analytical dual mesh method algorithm

The algorithm is similar to the dual mesh method algorithm de-scribed earlier. The method has the following steps:

1. The optimal coarse grid is generated using the vorticity-basedgridding of Ashjari et al. (2007, 2008) and the coarse-scale per-meability field is obtained using geometric averaging or anyother upscaling method that results into an isotropic perme-ability field.

2. At each time step, the saturation, permeability, and porositydata are given on the fine grid.

3. The average saturation for each coarse grid is calculated.4. For the current time step, the coarse-scale pressure equation is

solved numerically using the finite difference scheme thatyields the coarse-grid fluxes normal to the coarse-gridboundaries.

5. The local two-phase fine-scale pressure equation is solved ana-lytically over each coarse-grid block at each time step using Eqs.(21), (22). Incorporating Darcy’s Law, this reconstructs the fine-scale fluxes throughout the model.

6. Finally, the saturation field is updated from the reconstructedfine-scale velocity field.

In the next section, first we will evaluate our assumptions overthe staircase model. Next, the proposed method will be tested intwo challenging test cases. To investigate the effect of mobility ra-tio on our method, two mobility ratios are used in each test case.

5. Results and discussions

To evaluate the accuracy and effectiveness of our method, wewill test it on three test cases. In all the test cases, two mobility ra-tios i.e. M = 1 and M = 5 will be used where mobility ratio, M, is de-fined as mobility of the displacing phase (here the wetting phase)divided by mobility of the displaced phase (here the non-wettingphase) i.e. M = (krw,max/lw)/(krn,max/ln). Corey’s type relative per-meability is used to represent the wetting and non-wetting phaserelative permeabilities, i.e.:

kri ¼ kri;maxSi � Sir

Si;max � Sir

� �ni

i ¼ w;n ð23Þ

Here, we take krw,max = 0.6, krn,max = 1.0, Swr = Snr = 0.2, Sw,max =Sn,max = 0.8, nw = 1.5 and nn = 2. To achieve the aforementionedmobility ratios, the wetting phase viscosity is assumed 6 � 10�4 Pasand the non-wetting phase viscosity is assumed 1 � 10�3 Pas and5 � 10�3 Pas in the M = 1 and M = 5 cases, respectively. First, theintroduced method will be tested on the staircase method toevaluate the accuracy of our assumptions.

In all models, total simulation time is taken to be one pore vol-ume injected, i.e. tsim = VP/Qinj = 1PVI, where tsim is the total simula-tion time, VP is the pore volume and Qinj is the injection flow rate.In other words, 1PVI is the time needed for injection of one pore

200 D. Khoozan et al. / Journal of Hydrology 400 (2011) 195–205

volume of the displacing fluid. Simulations are performed on an In-tel E4500 CPU with 2 GB of RAM.

Fig. 4. Production curves for model 1, M = 1 case.

Fig. 5. Production curves for model 1, M = 5 case.

5.1. Model 1: the staircase model

This model is a synthetic model that its permeability field in logscale is shown in Fig. 3. As can be seen from the permeability field,the model has a base permeability of 1 mD (1mD = 10�3D = 9.86923 � 10�16 m2) and cells with permeability of10 and 100 mD form a staircase in the model. The dimensions ofthe model are 762 m long by 152.4 m wide by 7.62 m thick. Thefine-scale grid is 100 � 20 with uniform size for each of the gridblocks (dx = 7.62 m, dy = 7.62 m). Initially, the model is saturatedwith the non-wetting phase with 20% of residual wetting phase.Wetting phase is injected from an injection well located at the leftboundary of the model and the non-wetting phase is producedfrom a well on the right boundary.

To investigate the accuracy of the proposed assumptions, a5 � 5 coarse grid will be used. It is obvious that the best 5 � 5coarse grid would be a uniform one with DX = 152.4 m andDY = 30.48 m. Therefore, there is no need to perform the vortici-ty-based gridding technique. As it is clear from Fig. 3, using thiscoarse grid, the problem of equivalent permeability calculation willbe removed since all the cells in each coarse block have the samepermeability. In other words, the permeability field of the refinedgrid is the same as the permeability field of the fine grid. This prop-erty makes this model perfect for evaluating the assumptionsmade in our method since one of them i.e. performing the simula-tion on the refined grid is satisfied spontaneously. Hence, we canevaluate the accuracy of one of the most important assumptionsof our method, i.e. homogeneity of the saturation field over eachcoarse-grid block. To do so, we perform the simulation of the mod-el with the following methods:

1. Conventional simulation of the fine grid.2. Dual mesh method simulation on the refined grid.3. The proposed method (analytical dual mesh) simulation.

The production curve for different simulation methods areshown in Figs. 4 and 5 for M = 1 and M = 5 cases, respectively. Inthese curves, PVI is the pore volume injected of the wetting phasedefined as PVI = (Qinjt)/VP where Qinj is the injection flow rate of thewetting phase, t is the simulation time and VP is the pore volume ofthe model. Fo is the fractional flow of the non-wetting phase at theproduction well defined as Fo = Qn/(Qw + Qn) where Qw and Qn arethe flow rates of the wetting and non-wetting phases at the pro-duction well, respectively. As it is clear from the plots, the resultsof the dual mesh method and our method are approximately sim-ilar both in terms of breakthrough time and production history.The results of the proposed method and the fine grid simulationare also in great agreement for both cases. So it can be concludedthat the assumption of homogeneous saturation field is a reason-able one and not far from reality.

Fig. 3. Log scale permeability field of the staircase model in mD.

To investigate the accuracy of the saturation homogeneityassumption and hence the proposed method further, the saturationfield data will be compared. The wetting phase saturation fields ofthe different simulation methods at 0.4 PVI are shown in Figs. 6 and7 for M = 1 and M = 5 test cases. The results are similar to the pro-duction curve results. The saturation field of the proposed methodis approximately similar to the dual mesh method. The differencebetween the fine grid saturation field and the proposed methodis relatively small. As the saturation field results show, the accu-racy of our method compared to the dual mesh method and the re-fined grid simulations is relatively high. It can also be concludedthat the saturation homogeneity is a reasonable assumption.

After investigating the accuracy of the proposed method, thesimulation time will be compared. The simulation time of the dif-ferent methods are tabulated in Table 1. The results are exactly aswe expected. In both M = 1 and M = 5 cases, the simulation time ofthe proposed method is three times less than the fastest simulationmethod i.e. dual mesh method. It is also seven times less than thefine grid simulation time. So far, the results are in favor of our pro-posed method. Now the method will be examined on a realistictest case.

Fig. 6. Wetting phase saturation field of model 1 at 0.4 PVI, M = 1 case: (a) fine grid,(b) dual mesh method, (c) analytical dual mesh method.

Fig. 7. Wetting phase saturation field of model 1 at 0.4 PVI, M = 5 case: (a) fine grid,(b) dual mesh method, (c) analytical dual mesh method.

Table 1Simulation time of model 1 for different methods.

Simulation method Run time (s) M = 1 Run time (s) M = 5

Fine grid simulation 468 461Dual mesh method 201 200Analytical dual mesh method 64 69

Fig. 8. Log scale permeability field of the SPE 10 2D model in mD.

Fig. 9. The optimum coarse grid for model 2 overlaying on the vorticity map.

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5.2. Model 2: SPE10 2D model

This model is a modification of 10th SPE Comparative Project 2Dmodel (Christie and Blunt, 2001). The model is a two-phase modelthat has a simple 2D vertical cross-sectional geometry with no dip-ping or faults. The dimensions of the model are 762 m long by15.24 m wide by 7.62 m thick. The fine-scale grid is 100 � 20 withuniform size for each of the grid blocks (dx = 7.62 m, dy = 0.762 m).Initially the model is saturated with the non-wetting phase with20% of residual wetting phase.

The permeability distribution is a correlated geostatisticallygenerated field, shown in Fig. 8. The wetting phase is injected froman injection well located at the left boundary of the model and thenon-wetting phase is produced from a well on the right boundary.Both wells are completed vertically throughout the model.

To examine our method, first we have to generate a coarse gridbased on vorticity-based gridding technique. Here, an upscaling le-vel of 25 is considered, i.e. the number of blocks in the coarse gridis 25 times less than the number of cells in the fine grid. Imple-menting the vorticity-based technique described earlier, the16 � 5 coarse grid shown in Fig. 9 (the dashed lines) is found tobe the optimum coarse grid. In this figure, the normalized vorticityfield of the model is also shown. As can bee seen from the figure,the selected grid preserves the high vorticity regions and its gridboundaries are in the sharp vorticity change regions.

After selection of the coarse grid, we will investigate the speedand accuracy of our proposed method. To do so, we perform thesimulation of the model with the following methods:

1. Conventional simulation of the fine grid.2. Conventional simulation of the refined grid.

Fig. 10. Production curves for model 2, M = 1 case.

Fig. 11. Production curves for model 2, M = 5 case.

Fig. 12. Wetting phase saturation field of model 2 at 0.2 PVI, M = 1 case: (a) fine g

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3. Dual mesh method simulation on the refined grid.4. Analytical dual mesh simulation.

It should be mentioned that since the goal of our method isincreasing the simulation speed while reducing the numerical er-ror, the results should be compared to the refined grid. The differ-ence between results of the fine and refined grids are solely due tohomogenization error and is a measure of the accuracy of the vor-ticity-based gridding technique.

The production curve for the aforementioned simulation meth-ods are shown in Figs. 10 and 11. As it is clear, like the staircasemodel the results of the dual mesh method and our method areapproximately equal for both cases. The results of the proposedmethod and the refined grid simulation are also in great agree-ment. The little difference between the fine-grid model and othermethods is solely due to the homogenization error and is reportedto show the accuracy of the vorticity-based gridding.

The wetting phase saturation fields of the different simulationmethods at 0.2 PVI are shown in Figs. 12 and 13. The results aresimilar to the production curve results. The saturation field of theproposed method is approximately similar to the dual mesh meth-od. The difference between the saturation field of the refined gridand the proposed method is relatively small. The difference be-tween the fine grid results and other methods are again mainlydue to homogenization error. The simulation time of the methodsare tabulated in Table 2. The results are once again interesting. Thesimulation time of the proposed method is two times less than thedual mesh method. It is also 20 times less than the refined and finegrid simulation times.

From the obtained results, it can be concluded that the pro-posed method is accurate compared to dual mesh method and re-fined grid simulations. As a result, the assumptions of theanalytical dual mesh method seem to be reasonable. It is also givesa simulation speed boost with respect to the other methods. It canalso be observed that for both M = 1 and M = 5 cases, the conclu-sions are similar. Therefore, the method acts independently fromthe mobility ratio.

It is worth mentioning that since the speed difference betweenthe analytical and numerical solutions directly depends on thecoarse-grid block sizes and hence upscaling levels, it is expectedthat the speed-up factor with respect to the dual mesh method be-come higher for larger models and bigger upscaling levels. This willbe tested in the next challenging model.

rid, (b) refined grid, (c) dual mesh method, (d) analytical dual mesh method.

Fig. 13. Wetting phase saturation field of model 2 at 0.2 PVI, M = 5 case: (a) fine grid, (b) refined grid, (c) dual mesh method, (d) analytical dual mesh method.

Table 2Simulation time of model 2 for different methods.

Simulation method Run time (s) M = 1 Run time (s) M = 5

Fine grid simulation 1583 3306Refined grid simulation 1593 3320Dual mesh method 181 368Analytical dual mesh method 84 176

Fig. 14. Log scale permeability field of the model 3 in mD.

Fig. 15. The optimum coarse grid for model 3 overlaying on the vorticity map.

Fig. 16. Production curves for model 3, M = 1 case.

Fig. 17. Production curves for model 3, M = 5 case.

D. Khoozan et al. / Journal of Hydrology 400 (2011) 195–205 203

Fig. 18. Wetting phase saturation field of model 3 at 0.45 PVI, M = 1 case: (a) fine grid, (b) refined grid, (c) dual mesh method, (d) analytical dual mesh method.

Fig. 19. Wetting phase saturation field of model 3 at 0.45 PVI, M = 5 case: (a) fine grid, (b) refined grid, (c) dual mesh method, (d) analytical dual mesh method.

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5.3. Model 3: SPE10 3D model-layer 59

The last system considered is layer 59 from the three-dimen-sional model of Christie and Blunt (2001). This layer belongs tothe highly channelized region (lower portion i.e. layers 36–85) ofthe model. The permeability value in this model varies by above

six orders of magnitude, from a minimum of 0.0025 to a maximumof 20,000 mD. The Log scale permeability field of the model isshown in Fig. 14. The average porosity of the layer is used through-out the model (i.e. uavg = 0.1717). The fine-scale grid is a uniform220 � 60 grid with dx = 3.048 m and dy = 6.096 m. The initial con-dition and wells geometry are similar to the previous model.

Table 3Simulation time of model 3 for different methods.

Simulation method Run time (s) M = 1 Run time (s) M = 5

Fine grid simulation 95,033 91,392Refined grid simulation 34,323 33,714Dual mesh method 7465 7527Analytical dual mesh method 1707 1756

D. Khoozan et al. / Journal of Hydrology 400 (2011) 195–205 205

First, we should find the optimum coarse grid. In this model, anupscaling level of 100 is considered. The generated optimumcoarse grid based on the vorticity-based gridding technique isshown in Fig. 15 overlaying on the fine grid vorticity map. As canbee seen from the figure, the channels are captured by the vorticitymap and hopefully the obtained coarse grid will show the behaviorof these channels.

The utilized simulation methods are again similar to the previ-ous model. The production curve for these methods is shown inFigs. 16 and 17 for M = 1 and M = 5 cases, respectively. As can beseen from the figures, especially for the M = 1 case, the obtainedcoarse grid captured both channels at such a high upscaling level(i.e. 100). Again, like the previous models, the results of the dualmesh method and the proposed method are approximately equalfor both cases. The results of the proposed method and the refinedgrid simulation are also in good agreement but in less agreementwith respect to the previous models. This behavior can be ex-plained by the effect of the implemented high upscaling level. Athigher upscaling levels, the errors will be inevitably larger. Ascan be seen from the plots, the conventional dual mesh methodis also suffering from this problem.

Figs. 18 and 19 show the saturation field of the wetting phasefor different methods at 0.45 PVI. Here, the channels in fine andcoarse grids can be observed easily. As can be seen from these fig-ures, the proposed method captured these channels and the resultsare similar to the dual mesh method and refined grid simulation.The discontinuities on the saturation field of the analytical dualmesh method are due to its nature, since saturation field was as-sumed homogeneous over each coarse block. However, the resultsare satisfying overally. The difference between the fine grid resultsand other methods are again mainly due to homogenization error.

The simulation time of the methods are tabulated in Table 3.The results are more interesting here. The simulation speed ofthe proposed method is four times faster than the conventionaldual mesh method with almost the same accuracy, even in thiscomplicated model. It is also 20 times faster than the refined gridsimulation and 50 times faster than the fine grid simulation.

From the obtained results, once again it can be seen that theanalytical dual mesh method has the same accuracy as the dualmesh method and the refined grid simulation. It can also be con-cluded that the speed boost of the proposed method will becomemore in larger models with higher upscaling levels.

6. Conclusions

A novel simulation technique was presented that introducesanalytical solution into dual mesh method using realistic assump-tions. In this method, the coarse grid is generated using vorticity-based gridding technique because of its high accuracy. It is alsoused to make the applied assumptions more realistic.

The technique was applied to a synthetic model to evaluate theimplemented assumptions and they were found to be realistic. Themethod was also tested on two realistic models to investigate itsaccuracy and speed. Since the aim of the method is mainly reduc-ing the numerical errors while increasing the simulation speed, the

results of the refined grid simulation were taken as the referencesolution. The results showed that the proposed method and thedual mesh method yielded approximately similar results. Theaccuracy of the proposed method with respect to refined grid sim-ulation was also quite high.

In terms of simulation time, the proposed method was 2–4times faster than the dual mesh method (with the same simulationresults) and 7–20 times faster than the refined grid simulationtime (with a high accuracy). It was also concluded that for largermodels with higher upscaling levels, the speed-up factor with re-spect to the dual mesh method would become higher.

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