generation of voronoi grid based on vorticity for coarse-scale modeling of flow in heterogeneous...

Transp Porous Med (2010) 83:541–572DOI 10.1007/s11242-009-9458-2

Generation of Voronoi Grid Based on Vorticityfor Coarse-Scale Modeling of Flow in HeterogeneousFormations

M. Evazi · H. Mahani

Received: 15 April 2008 / Accepted: 6 August 2009 / Published online: 9 October 2009© Springer Science+Business Media B.V. 2009

Abstract We present a novel unstructured coarse grid generation technique based onvorticity for upscaling two-phase flow in permeable media. In the technique, the finenessof the gridblocks throughout the domain is determined by vorticity distribution such thatwhere the larger is the vorticity at a region, the finer are the gridblocks at that region. Vor-ticity is obtained from single-phase flow on original fine grid, and is utilized to generate abackground grid which stores spacing parameter, and is used to steer generation of triangularand finally Voronoi grids. This technique is applied to two channelized and heterogeneousmodels and two-phase flow simulations are performed on the generated coarse grids and, theresults are compared with the ones of fine scale grid and uniformly gridded coarse models.The results show a close match of unstructured coarse grid flow results with those of finegrid, and substantial accuracy compared to uniformly gridded coarse grid model.

Keywords Upscaling · Vorticity-based gridding · Unstructured grid · Heterogeneousporous media · Coarse grid · Two-phase flow

List of SymbolsA External area of control volumeAi Area of control volume idmax Maximum spacing parameterdmin Minimum spacing parameterf Fluid cutE Error

M. EvaziDepartment of Chemical and Petroleum Engineering, Sharif University of Technology, Tehran, Iran

H. Mahani (B)Shell International Exploration & Production B.V., Kessler Park 1, Rijswijk 2288 GS, The Netherlandse-mail: [email protected]

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542 M. Evazi, H. Mahani

G Source term�g Gravity accelerationh Gridblock length�i , �j , �k Unit vectorsK Permeability tensork Permeabilitykr Phase relative permeabilityk̄ Upscaled permeabilityL Spacing parameterM Number of block edgesN Number of source points�n Outward normal unit vectorP Fluid pressureqp Well flow of phase pS Fluid saturationt Time�u Velocity vector (�u = ux�i + uy �j + uz �k)UL Upscaling levelV Control volume

Greek Symbolsβ Relaxation factor� Phase potentialϕ Porosityλ Mobilityμ Fluid viscosityρ Fluid density�ω Vorticity vectorψ Intensity of source element

Superscriptsis Internal source elementr Iteration numbert Time level

Subscriptsc Coarsef Finen nth Sourceo Oilp PhaseT Totalw Waterwf Well flowing in Pwf

x x directiony y directionz z direction

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Generation of Voronoi Grid Based on Vorticity in Heterogeneous Formations 543

1 Introduction

Simulation of flow and transport through permeable media and underground formations,i.e., hydrocarbon reservoirs and aquifers, involves introduction of geologically realistic mod-els, which describe petrophysical properties such as permeability. With fast advancement ofreservoir characterization technologies, these data are becoming more and more detailedwhile advances in processing technologies fail to counterbalance with. This high level ofdetails makes the use of fine geological models known as fine grid for reservoir simula-tion and management, time consuming, and computationally expensive. This has led to theevolution of upscaling techniques which attempt to decrease the number of the numericalgridblocks used for storing the data and solving flow equations.

Upscaling is the replacement of a finely described heterogeneous model with coarsermodel, and involves calculation and assignment of homogenized properties to the coarsegridblocks based on the underlying fine grid values. There are various upscaling techniquesin literature that can be employed for this purpose, the choice of which depends on therequired accuracy and the underlying heterogeneity (Wen and Gómez-Hernández 1996a,b;Renard and de Marsily 1997; Farmer 2002; Durlofsky 2003). Performance of the upscaled(coarse grid) model not only is affected by the selected upscaling technique, but also is influ-enced by coarse grid size and distribution i.e., homogenization volume (Durlofsky 2003; Wenet al. 2003a,b); therefore upscaling is often combined with optimized gridding techniques.

Coarse grid generation from finely gridded geological model has been subject of activeresearch and still remains a challenging area today. The task of a gridding technique is opti-mizing size, number, and location of gridblocks by identifying the critical regions where thegridblocks should remain fine, and the regions where the blocks are coarsened. In order toensure that the coarse grid model flow results can closely match those of fine grid, the pro-cess should also preserve key reservoir heterogeneities and their connectivity. Furthermore,coarse grid generation should be managed such that coarse gridblocks with small variationof rock and flow properties are produced. Upscaling of rock absolute permeability wouldbe also less challenging when the underlying permeability is rather homogeneous (Christie1996; Barker and Thibeau 1997; Renard and de Marsily 1997; Durlofsky 2003).

So far, several methods have been developed which utilizes some kind of geologic orflow properties to produce the coarse grid. Some of these simply use the permeability varia-tion as the criterion for coarsening, as they try to minimize the permeability variation in thegenerated grid (Garcia et al. 1992; Farmer et al. 1991; Li et al. 1995; Qi et al. 2001). Themethod proposed by Garcia et al. (1992) uses the concept of elastic grid and involves theadjustment of block boundaries of an initial uniform coarse grid to reach an optimum gridwith minimum permeability variation. In the grid generated, the higher is the permeabilityvariation in an area, the finer would be the gridblocks at that area. In this regard, global per-meability variation has been incorporated in the grid generation too (Li et al. 1995). Thesetechniques are acceptable, as they do not require flow simulation on the fine grid and thus arecomputationally fast. However, these techniques are basically static and do not necessarilyrespect the flow connectivity, as flow pattern and different well locations in a given domaindo not alter the grid.

Some other methods use flow information to obtain a coarse grid (Durlofsky et al. 1996,1997; Darman et al. 2000; Castellini 2001). These methods entail solution of the single-phaseflow in the fine grid, and generation of the streamlines and equi-potential lines. These linesare orthogonal and form the gridblocks. In the grid generated, the denser are the streamlinesat a region, the finer are the gridblocks at that region. These techniques highly respect the flowcondition and usually predict well the fluid displacements. However, although flow-based

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techniques preserve high flow paths and total fine grid flow in the coarse grid, there is noguarantee that permeability distribution and its variation between layers or inside geologicalunits are preserved. This is opposed to permeability-based gridding methods. The lack ofexplicit investigation of fine grid permeability and its variation within the blocks can leadto generation of coarse gridblocks with high permeability variations, especially in the lowerflow areas where coarse gridblocks are made larger. This not only introduces inaccuraciesin coarse grid model due to loss of fine grid permeability information, but also challengesconventional permeability upscaling techniques that cannot capture properly variation (dis-tribution) of permeability (or Darcy velocity) in the coarse blocks, and often produce diagonalupscaled permeability tensor, i.e., considering only “translational” movement of flow whileneglecting fluid rotation (Mahani et al. 2009). In addition, while the flow-based griddingleads to very large coarse blocks in low flow regions, it introduces unnecessary refinementsin high flow, homogeneous layers (corresponding to irrotational flow regions). This additionalrefinement does not improve homogenization or upscaling process (which is the underlyingidea of grid generation techniques), while it can only be justified to control mostly numeri-cal dispersion (see, Fincham et al. 2004 for field application of flow-based techniques, andMostaghimi et al. 2009 for comparison of gridding techniques).

There are other techniques, which combine flow and permeability information, with logi-cal expectation of generating a better grid. Stern and Dawson (1999) proposed an algorithmthat uses both flow and permeability variation to determine the optimum coarse grid. Wenand Gómez-Hernández (1996a,b, 1998) introduced the idea of selective iterative upscalingthat uses both permeability and velocity as variable to generate coarse grid. Initially, themethod constructs a coarse grid only based on permeability field, using the elastic grid con-cept as proposed by Garcia et al. (1992), and then iteratively adjusts the grid based on flowvelocity interpolated from the coarse grid solution. The selective gridding produced resultswith greater accuracy compared with permeability-based gridding due to better preserva-tion of high flow channels. The main drawback of the technique is the construction of asuitable initial coarse grid which depends on the applied field boundary conditions. In addi-tion, the procedure entails two separate steps and consequently is time consuming. EllipticJacobian-based technique was also proposed by He (2004) and He and Durlofsky (2006),which involves deriving the solution of a set of nonlinear elliptic equations and obtaining atransformation to map from physical space to logical Cartesian one. These transformationsattempt to preserve different information from the fine scale description of the domain such asflow and geological information. Jacobian-based grid generation is fast and simple to apply.However, in some cases, the convergence is slow and there is no guarantee for unfolding ofthe grids (Durlofsky 2005).

Fluid vorticity from single-phase flow, which is the cross product of velocity and perme-ability gradient, has been recently proposed to generate non-uniform coarse grid by Mahani(2005), Mahani and Muggeridge (2005), Ashjari et al. (2007), Mahani et al. (2007), Ashjariet al. (2008), and Ashjari et al. (2009). Vorticity by definition incorporates both permeabilityvariation and flow condition in the grid generation, and is attractive as it captures both mea-sures in a single quantity. In this technique, the larger is the vorticity at a region, the finer arethe gridblocks at that region.

The success and advantages of the so-called vorticity-based technique has been shownon the Cartesian grid; however, the inherent inflexibility of Cartesian gridblocks does notallow complete exhibition of the method’s capabilities. Furthermore, inherent lack of con-trol on cell sizes, difficulty to conform to internal and external boundaries, and the need forlarge number of gridblocks for handling of complex geometry reservoirs, all restrict the useof structured grids. All these have motivated the researchers to move towards unstructured

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grids which have proved to help reduce the grid orientation effect, preserve the permeabilityheterogeneities, provide the capability to specify grid points anywhere inside the domainregardless of the location of other points, and conform to the internal and external bound-aries. Several types of unstructured grid have been utilized in reservoir simulation, a goodreview of which can be found in Aziz (1993). In this study, the Voronoi unstructured gridwhich is among the most promising ones are used for the generation of vorticity-basedunstructured grid.

This article is laid out as follows. In the next sections, first vorticity concept and its cal-culation on structured and unstructured grids are discussed. Next, the proposed algorithmto generate coarse unstructured grid is presented. Two highly heterogeneous models willbe used to test the capabilities of the method in generating unstructured coarse grid. Thentwo-phase flow results of the generated coarse grids are compared with those of fine grid.Finally, discussion and concluding remarks are presented.

2 Vorticity Concept

Vorticity was first used by Josselin de Jong (1960) for modeling of multi-phase porous mediaflow and then used for the quantification of permeability heterogeneities by Chen and Meiburg(1998). The idea of using vorticity in coarse grid generation from a finely gridded geomodelwas first proposed by Mahani and Muggeridge (2005) and Mahani et al. (2009).

Vorticity is of special interest in fluid dynamics. Vorticity is a vector describing rate anddirection of rotation of fluid element at any given location. Flow in porous media and res-ervoirs is characterized by vorticity, which is the result of local heterogeneities, and flownon-uniformity (Josselin de Jong 1960; Bear 1972; Sposito 1994, 2001; Mahani 2005). Insimulation of flow through porous media, vorticity is practically ignored; however, in non-porous media it is extensively used in combination with Navier–Stokes equations and isdescribed by conservation equations. For flow in porous media, vorticity is not conserved inthe same sense as in Navier–Stokes flow, because we cannot define an infinitesimal controlvolume and then analyze the various fluxes of vorticity into and out of this volume to set up apartial differential equation (PDE) (Mahani 2005). Therefore, we have an algebraic equationrelating vorticity intensity to instantaneous properties such as velocity, fluid distribution, vis-cosity, and density fields. In order to derive vorticity equation, we start with its mathematicaldescription.

Vorticity, by definition, is related to the curl of velocity field,

�ω = �∇ × �u = �i(∂uz

∂y− ∂uy

∂z

)+ �j

(∂ux

∂z− ∂uz

∂x

)+ �k

(∂uy

∂x− ∂ux

∂y

). (1)

Velocity of flow in permeable media can be obtained from extension of Darcy’s law totwo-phase flow,

�uT = −λT K �∇�; λT = λo + λw, (2)

where λ denotes the mobility of fluid (fluid relative permeability divided by viscosity) and isa function of saturation in pore space, K is the absolute permeability tensor and� is the flowpotential. Subscripts T, o, and w refer to the total, displaced fluid (here oil), and displacingfluid (here water), respectively.

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By substituting the velocity Eq. 2 in Eq. 1, general vorticity equation for two-phase flowin 3D space is derived (Mahani 2005; Mahani et al. 2009),

�ω = −�uT × �∇ ln K − 1

λT

dλT

dSw�uT × �∇Sw −ρKλT

d fw

dSw�g × �∇Sw, (3)

in which, S, ρ, f , and g refer to the fluid saturation, fluid density, phase fractional flow, andacceleration due to gravity. In the above derivation, capillary pressure is ignored, and for sim-plicity, it is assumed that permeability is isotropic and that the principal axes of permeabilitytensor are aligned with main flow directions i.e., off-diagonal terms are zero.

It can be seen that vorticity, in this case, is given by the sum of three terms: a term depen-dent on permeability variations (the heterogeneity term), a term dependent on changes intotal mobility and the saturation gradient (the mobility term), and a term dependent upon thedensity difference, changes in the fractional flow with saturation, and the saturation gradient(the gravity term). The relative magnitude and importance of these terms depend essentiallyon the flow regime (viscous to gravity ratio), mobility ratio, and stability of displacement,type, and distribution of heterogeneity. Heterogeneity term is a maximum when there is largepermeability variation perpendicular to flow. For example, for flow parallel with layered for-mations, the vorticity is largest around boundaries of layers. The mobility term represents thecontribution of fluid mobilities and saturation to vorticity. It is significant when saturationgradient is perpendicular to flow direction. However, in most of the displacements occur-ring in reservoir, the largest saturation gradient exists in flow direction except when thereis layering, channeling, and fingering. Channeling occurs mostly due to high permeabilitystreaks and high permeability contrast, and fingering is usually triggered by presence oflocal heterogeneities. However, in these cases, the heterogeneity term will also be large. Thegravity term will be large when saturation gradient is perpendicular to gravity direction. Thisoccurs mostly for viscous-dominated displacements and for these cases, density differencewill be small, and gravity term will not be that much considerable. Therefore, in most of theconditions prevailing in field application, heterogeneity term is the most controlling factoron vorticity as it exists even in single-phase flow, and it is less affected by the variation oftwo-phase flow properties such as saturation in the course of the simulation (see Mahani et al.2009 for further details). Therefore, vorticity of two-phase flow can be closely approximatedusing single-phase vorticity. As a result, only single-phase flow simulation is performed onthe original fine grid to calculate vorticity.

2.1 Vorticity Calculation

We apply the vorticity-based unstructured gridding technique to 2D areal (x − y plane) het-erogeneous models for which vorticity vector has only one non-zero component, ωz , in the zdirection perpendicular to flow plane. No vorticity is generated in flow direction. Thereforeonly ωz should be calculated.

Calculation of vorticity on the fine grid requires solution of pressure equation:

∇ ·(

K �∇ P)

= 0, (4)

in which, K is the permeability tensor, and P is the fluid pressure. The actual field boundaryconditions are used to solve the above equation numerically on the fine grid except that onlysingle-phase flow is modeled. From numerical solution of Eq. 4, the pressure distributionand then fine grid velocity field are computed.

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1,

2y i j

u+

i,j 1

,2

y i ju

−

1,

2x i j

u−

1,

2x i j

u+

Averaging area

(a) (b)

Fig. 1 a Velocities at block boundaries used to calculate vorticity. b Four neighboring velocities used toestimate velocity at block boundaries (Mahani 2005; Mahani et al. 2009)

2.1.1 Structured Cartesian Grid

For Cartesian grid, velocity is substituted in the descretized from of Eq. 1 following approxi-mation of first derivatives by first-order central finite difference schemes to yield the vorticityequation:

ωz(i, j) = ∂uy

∂x

∣∣i, j − ∂ux

∂y

∣∣i, j

∼=(uy

(i + 1

2 , j) − uy

(i − 1

2 , j))

x−

(ux

(i, j + 1

2

) − ux(i, j − 1

2

))y

. (5)

In this equation, (i , j) denotes the location of the gridblock, x and y are the lengths ofgridblock in x and y direction, and ux and uy refer to face-centre velocities. Velocity termsin Eq. 5 cannot be calculated directly, thus these are estimated from the appropriate valuesof the neighboring velocities (Fig. 1) as follows:

ux

(i, j ± 1

2

)

∼= 1

4

[ux

(i − 1

2, j ± 1

)+ ux

(i + 1

2, j ± 1

)+ ux

(i − 1

2, j

)+ ux

(i + 1

2, j

)]

uy

(i ± 1

2, j

)

∼= 1

4

[uy

(i ± 1, j − 1

2

)+ uy

(i ± 1, j + 1

2

)+ uy

(i, j − 1

2

)+ uy

(i, j + 1

2

)]. (6)

By replacing terms of Eq. 5 with relevant ones from Eqs. 6, vorticity can be calculated forevery gridblock and is assigned to the gridblock centre.

2.1.2 Unstructured Grid

In unstructured grid, we consider the integral form of the vorticity equation. This integralis also called circulation and is measure of rotation in control volume. Assuming 2D flow

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Fig. 2 Control volume i and itsneighbor j

i

j2

1

k

through control volume (i) with surface area Ai , shown in Fig. 2, we obtain

∫Ai

ωzdxdy =∫Ai

(∂uy

∂x− ∂ux

∂y

)dxdy. (7)

The left-side of Eq. 7 can be written as

∫Ai

ωzdxdy = ω̄z Ai , (8)

where ω̄z is the average vorticity in control volume. On applying Green’s theorem to theright-hand side of Eq. 7 and approximating the resulting integrals, we obtain

ω̄z Ai =∫Ai

(∂uy

∂x− ∂ux

∂y

)dxdy =

∮(uydy + ux dx) =

M∑j=1

(uy,ky12 + ux,kx12

)(9)

In this equation, M is the number of neighbors of block i . The values of uy and ux at pointk, the intersection of connection ij with edge 12 (see Fig. 2), are estimated as

uy,k = 1

Ai1 j2

∫i1 j2

(ky

μ

∂P

∂y

)dxdy

= − 1

Ai1 j2

((ky

μ

)i

Pi + P1

2xi1 +

(ky

μ

)j

P1 + Pj

2x1 j

)

− 1

Ai1 j2

((ky

μ

)j

Pj + P2

2x j2 +

(ky

μ

)i

P2 + Pi

2x2i

). (10)

ux,k can be similarly approximated. Pressure values at points 1 and 2 on the edges can beaveraged from the surrounding nodes.

Finite volume discretization of vorticity equation is only preferable to use in complexmodels, where Cartesian grids fail to correctly represent the features. However, structured(Cartesian) discretization of vorticity equation is simpler and more efficient on fine grid, andwill be used in the models being investigated in this article.

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3 Coarse Grid Generation Based on Vorticity

Application of vorticity as criterion for coarse grid generation and grid size adaptation wasfirst presented by Mahani and Muggeridge (2005). The technique was improved by Ashjari etal. (2007), Mahani et al. (2007), and Ashjari et al. (2008). In this technique, the block bound-aries are manipulated according to the vorticity intensity to optimize the homogenization offinely gridded geological models. The approach entails a single-phase flow simulation onthe fine scale model. This provides velocity field from which vorticity is calculated. Then, acoarse grid is generated by minimizing the difference between fine and coarse grid vorticitydistributions for a certain upscaling level. This has been shown to perform well in recognizingthe critical heterogeneities and flow areas. However, the original technique was developed inCartesian grid, which suffers from inherent inflexibility to capture complex geometries andchannel boundaries and makes the local grid refinement difficult. Use of unstructured gridimproves the coarse grid generation based on vorticity, as it has greater flexibility and highlyenables the local refinements.

3.1 Proposed Unstructured Coarse Grid Generation Algorithm

As already mentioned, use of unstructured grid makes the subdivision of physical domainhighly flexible so that it is possible to generate grids which contain gridblocks with varyingsize and shape. This process provides grids with controllable resolution according to the localcriticalities. Inspired from this, we anticipate improvement over former structured (orthog-onal) vorticity-based grid generation technique (presented in Mahani and Muggeridge 2005and Mahani et al. (2009)) on the use of unstructured grid. The following steps are used inthis study:

1. Building/importing fine scale geological model as the primary input2. Calculation of vorticity on the fine scale model as described in Sect. 2.13. Construction of background grid

The idea of background grid was first used by Pirzadeh (1993) for the control of Advanc-ing Front Triangulation (AFT) for problem concerned in aerodynamics (non-porous mediaand open spaces). However, the technique did not mean to generate coarse grid, rather onlya computational unstructured grid. The technique uses the concept of spacing parameter,which plays an essential role in determination of grid point density, and thus block sizes. Forproblems in Geosciences, spacing parameter, which is a measure of distance between gridpoints, can be derived from (a suitable fine grid) physical quantity important in flow dynam-ics and computation. Note that background grid is different from fine grid (or geologicalmodel) in a sense that fine grid describes porous media’s static and dynamic properties suchas distribution of permeability and porosity, while background grid only stores informationabout grid properties such as grid point density and spacing parameter. The sole function ofbackground grid is for interpolation of grid geometry characteristics and to guide the AFT ininsertion of new grid points at suitable locations for generation of a coarse grid. A fine gridgeological model is primary in the creation of a background grid.

In this article, we use single-phase flow vorticity, a measure for adjustment of grid points,in other words, for the calculation of spacing parameter and grid point distribution. In orderto construct the background grid, a process, similar to solving heat equation in a medium withinternal heat sources, is used. This process can be described by Poisson’s equation, whichinstead of being solved for temperature or isotherms, is solved for spacing parameter in thephysical domain i.e., the fine grid.

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In order to control the distribution of spacing parameter, a number of source elements areintroduced in the domain, which locally and directionally determine the spacing or grid pointdensity. The sources can be internal or external. Internal sources control spacing inside thedomain and are determined by the intensity of vorticity at each location. In order to define theinternal sources, a cut-off (threshold) value is specified for the vorticity. Grid cells at whichvorticity is above the cut-off are selected as internal sources. The internal sources act also asinner boundary conditions to the Poisson’s equation. A spacing parameter and an intensity(strength) factor are assigned to each source. Values of these two parameters are directlydependent on the vorticity and the required upscaling ratio. We employ a linear relationshipfor the spacing parameter of internal sources (L is), as follows:

L is = dmax + |ωnor| (dmin − dmax), (11)

in which, ωnor, dmin, and dmax are the normalized vorticity, minimum and maximum sizeof spacing parameter, respectively. Normalized vorticity is simply obtained by dividing thevorticity values by the absolute value of maximum vorticity found in the fine grid. dmin isthe desired minimum size of coarse grid blocks (that can be equal to fine gridblock size),while dmax is determined from upscaling level (number of gridblocks in fine grid to the onein coarse grid). dmax defines the maximum size of coarse gridblocks. Setting vorticity cut-offto zero will result in the original fine grid, while cut-off value of one yields a uniform coarsegrid for the specified upscaling level.

External sources are introduced by the user to specify the spacing at the boundaries.Uniform spacing with value of dmax can be used for spacing parameter at the boundaries,unless otherwise is necessary. This determines the outer boundary condition to the Poisson’sequation. Determination of intensity factor (strength) of all the source elements can be madeby trial and error such that a grid with specified number of gridblocks is obtained.

Once intensity factor and spacing parameter of all the source elements are assigned, spac-ing parameter in the rest of domain can be modeled by solving the Poisson’s equation asfollows:

∇2 L = G, (12)

where L is the spacing parameter and G is the sum of internal source terms. The inner andouter Dirichlet boundary conditions to this equation were explained above. The equationis descretized on the fine grid and is solved iteratively by implementation of Gauss–Seidelscheme with successive over-relaxation to yield the following (Pirzadeh 1993):

Lr+1i, j = (1 − β)Lr

i, j + β

(Lr+1

i, j−1 + Lri, j+1 + Lr+1

i−1, j + Lri+1, j + h2

N∑n=1

ψn In

)

/(4 + h2

N∑n=1

ψn Jn

), (13)

where,

In = Ln

r2n, Jn = 1

r2n. (14)

In these equations, ψn , Ln , rn , β, h, and N are intensity factors of sources which aresubstantial part of G, spacing parameter of internal sources, distance between point (i, j)and nth source, relaxation factor, gridblock length, and number of the source elements,respectively. Superscript r refers to iteration level. Due to the diagonal dominance of the

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Fig. 3 Calculation of effectivepermeability for connection i– j

coefficient matrix, this iterative solution is stable. The convergence rate of this process canbe accelerated with an optimum value of β. For some cases of special geometries, theoreticaloptimum values are available which can be found in Lapidus and Pinder (1982).

4. Application of advancing front triangulation, Delaunay tessellation, and Voronoi gridgeneration

In advancing front triangulation, boundary of the domain forms the initial front, and a newtriangle is developed on the smallest edge of the front. The third vertex of the triangle isdetermined by the spacing parameter value at the midpoint of the edge which is interpolatedfrom the background grid. Front advances in the closed domain of interest until front con-tains no edge. Details of this method can be found in Farrashkhalvat and Miles (2003). Inorder to obtain the Voronoi or Perpendicular Bisector (PEBI) grid, Delaunay tessellation isperformed on the grid points generated by advancing front. Delaunay forms the triangles insuch a way that the circle circumscribing vertices of each triangle does not contain any of theother grid points. The triangular grid generated by Delaunay is of higher quality, compared tothe one generated by advancing front method. Voronoi gridblock vertices are obtained fromthe center of circles circumscribing the triangle’s vertices.

Steps 3 and 4 are not fully automatic, as we need to follow some trial and error procedureon spacing parameters at source elements, location of external elements, and intensity factorof the source elements to get the grids with specified upscaling level. Thus, these two stepsare repeated until a grid with the intended upscaling level is obtained.

5. Allocation of upscaled permeability to the coarse gridblock

In coarse-scale flow modeling, the coarse grid generation and upscaling calculation are twocomplementary components. The coarse-scale grid would be at most as good as the follow-ing upscaling calculation and discretization technique. Nonetheless, as the main purpose ofthis article is development of unstructured grid based on vorticity, the impact of upscalingtechnique is not investigated here. In this study, a simple analytical (arithmetic–harmonic)method is used to upscale fine grid permeability. Similar method was used by Palagi (1992).

As shown in Fig. 3, permeability of connection i − j , k̄i j , is determined by specifying somepoints on two triangles between the grid points i and j , with their values being interpolatedfrom the fine scale model. Simple arithmetic and harmonic averaging is applied as below tocalculate the effective permeability,

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k1−3 =(

3∑l=1

kl

) /3, k4−6 =

(6∑

l=4

kl

) /3, k7−9 =

(9∑

l=7

kl

) /3, (15)

k̄i j = 31

k1−3+ 1

k4−6+ 1

k7−9

. (16)

This method is used here because of its simplicity, easiness of implementation, fastness,and the ability to produce coarse-grid flow results with fairly good agreement with those offine grid. However, it suffers from some disadvantages as it provides an upper bound forthe upscaled permeability values and simply assumes that flow within the coarse gridblockis parallel with the connection e.g., i– j at all the points. It, therefore, neglects the actualflow behavior and true boundary condition at each coarse grid unlike flow-based upscalingtechniques e.g., pressure-solver (Begg et al. 1989). These weaknesses make this method lessaccurate than numerical flow-based upscaling techniques. This implies use of more realisticmethods such as pressure-solver methods, flux-boundary conditions (Wallstrom et al. 2002),and upscaling of transmissibility instead of the permeability, although implementation ofthese techniques, which may entail full tensor upscaling, is numerically intensive in unstruc-tured grid. Nevertheless, it is assumed that method of upscaling will have equal impact onthe results of uniform grid and non-uniform unstructured grid.

4 Domain of Method Applicability

Vorticity-based gridding works essentially in heterogeneous formations because vorticity isa measure of heterogeneities, according to Eq. 3. Vorticity is influenced by permeabilityvariation and flow field, and indicates relative importance of heterogeneities on the flow.The technique in essence aims at effective homogenization of heterogeneous systems byrecognizing areas with high permeability variation perpendicular to flow and refining (orretaining) those areas while, coarsening the areas with a small degree of heterogeneity orin effect homogeneous areas. Therefore, the technique does not suggest any refinement inhomogeneous areas even if they are situated in high flow regions as in channels or aroundwells, although the method preserves connectivity. In these cases, what mostly controls theaccuracy of flow results is the numerical dispersion resulting from inefficiency of numericalsolution procedures. In order to minimize numerical dispersion, proper capturing of fluid–fluid interface or saturation front is mandatory. This can be achieved by using high-order(second- or higher-) numerical techniques, improved fluid mobility calculations, multi-scaleupscaling techniques (Mahani et al. 2007; Ashjari et al. 2009), or conventional local gridrefinement e.g., around wells. In order to compliment the vorticity-based gridding algorithmin such cases, a modular gridding approach can be adopted whereby various gridding capa-bilities are used to tackle both homogenization and numerical dispersion issues. For instance,a grid refinement module can be deployed in the vicinity of domain boundaries and naturalelements such as injector-producers, and another gridding module based on vorticity in thevicinity of medium-fluid heterogeneity. This would lead to a more optimal grid in such cases.

5 Governing Flow Equations

We consider viscous-dominated two-phase flow in heterogeneous porous media. The perme-ability heterogeneity is assumed to be the main controlling factor on fluid movement. Flow

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simulations are performed without considering fluid compressibility, gravity, and capillaryeffects. The two phases are designated water (subscript, w) and oil (subscript, o). Flow equa-tion for each phase is formulated by combining mass conservation equation and extensionof Darcy’s law for two-phase flow.

Mass conservation equation for each phase (p) in a control volume, V , with an externalsurface area A is:

∮A

λP K �∇ P · �ndA = ∂

∂t

∮V

SpφdV + qp, (17)

where subscript p denotes the phase, t time, φ porosity, and P the fluid pressure. λP is thephase mobility defined by the phase relative permeability (krP(S)) divided by the fluid vis-cosity (μP), S is the water saturation, �n is the outward normal unit vector to surface A, andthe source term qp is the well flow of phase p.

We set up flow equations (pressure and saturation) for each control volume and applytwo-point stencils for flux approximation (TPFA) suitable for PEBI-type grid in the absenceof permeability anisotropy. We employ standard IMPES Control Volume Finite Differenceformulation to solve flow equations. The flow equations on fine and coarse grid are essentiallyof the same form, except that, in the coarse grid, we use the upscaled permeability instead offine grid permeability.

6 Numerical Results

The proposed algorithm is used to subdivide the physical domain of two highly heteroge-neous models: Layer 37 from Model 2 of 10th SPE Comparative Solution Project (Christieand Blunt 2001), and a synthesized, channelized model. These models are very challengingfor upscaling and require efficient gridding. In order to evaluate the grids generated by thealgorithm, two-phase water/oil displacement is simulated in these models, where water asdisplacing fluid is injected to displace oil in fully oil-saturated rock. Line-drive and point-drive injection/production schemes are considered to study the impact of flow scenario on thegrid structure and flow performance. In the line-drive scheme, water is injected from left side,and oil is produced from the right side. Both injector and producers are completed across thewhole model thickness. In the point-drive scheme, injection/production is limited at certaininjection/production points inside the model. No flow boundary conditions are imposed atthe top and bottom of the model in case of line-drive. However, in point-drive scheme, all themodel boundaries are sealed. Gravity and capillarity effects on flow are neglected. Two-phaseflow simulations are first performed for unit–mobility ratio displacement, then the impactof non-unit–mobility ratio on the performance of generated coarse grids, and their processdependency is investigated.

In order to quantify the relative performance of the generated coarse grids, time averageoil production rate error (ETAOPR, Eq. 18) and breakthrough time error (EBTH, Eq. 19) arecalculated from fine and coarse grid two-phase results,

ETAOPR =∑

EOPR(i)t (i)∑t (i)

, (18)

EBTH = |BTHf − BTHc|BTHf

, (19)

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Fig. 4 Log permeability distribution of layer 37 from SPE 10th CSP Model 2

where

EOPR = |Qof (i)− Qoc(i)|Qof (i)

. (20)

In these equations, Qof , Qoc, BTHf , BTHc, and i refer to oil production rate computed byfine grid, oil production rate computed by coarse grid, breakthrough time obtained from finegrid solution, breakthrough time obtained from coarse grid solution, and the sampling time,respectively.

6.1 Model One: Layer 37 from Model 2 of 10th Comparative Solution Project (CSP)

This model is selected for the evaluation of our proposed algorithm. The model is com-plex, highly channelized, and heterogeneous where the permeability in the system undergoesa wide variation from 0.004 to 20,000 mD. Fine grid permeability distribution on regularCartesian grid is shown in Fig. 4. Fine grid resolution is 220×60.

The procedure proposed above is applied to this case to generate an unstructured coarsegrid with upscaling level of almost 17. Vorticity map, background grid, and Delaunay andVoronoi grids for this model are shown in Figs. 5, 6, 7, and 8. As expected from permeabilityvariation, vorticity is widely distributed in the model that results in a background grid ofvarying size for spacing parameter. Hence, a promising non-uniform coarse grid is generatedin which the grid point density is smoothly varying from high- to low-density values. Table 1summarizes the determining parameters used to generate the background, triangular, andVoronoi grids.

Table 1 Parameters used incoarse grid generation for SPE10th CSP Model 2, layer 37

Normalizedvorticity cut-off

dmin dmax ψmin ψmax

0.5 20 ft 80 ft 0.0385 20

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Fig. 5 Vorticity map of layer 37 from SPE 10th CSP Model 2

Fig. 6 Background grid generated for SPE 10th CSP Model 2, layer 37: vorticity cut-off = 0.5

Two-phase flow is simulated on the generated coarse grid (see Table 2 for relevant sim-ulation data) and compared with the results of fine grid, and of a uniform (unstructured)coarse grid with the same number of gridblocks as our non-uniform unstructured coarse grid(Fig. 9). The figure shows a promising match of the fine grid oil recovery by the non-uniformcoarse grid generated from the proposed method, while uniform coarse grid fails to predictfine scale result. Time average oil production rate error of the uniform and the unstructuredcoarse grids are 57% and 8.6%, respectively, and breakthrough time is calculated with 60%and 3.3% error, respectively.

Table 2 Relevant data for SPE10th CSP Model 2, layer 37

Porosity μo μw kro krw Qinj Pwf atproductionwell

0.1717 1 cp 1 cp S2o S2

w 500 STB/day 3,000 psi

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Fig. 7 Unstructured Delaunay triangular grid for SPE 10th CSP Model 2, layer 37

Fig. 8 Non-uniform unstructured Voronoi grid for SPE 10th CSP Model 2, layer 37

In order to evaluate the effect of upscaling level on the quality of results, a coarse gridmodel with upscaling level of 35 was generated. In Figs. 9 and 10, the two-phase flowperformance of this model is compared with those of (non-uniform) unstructured coarsegrid and the uniform grid with upscaling level of 17. These results suggest that grids withdifferent upscaling level will exhibit varying performance as expected, such that with thedecrease of upscaling level unstructured grids will perform better. It should be also notedthat non-uniform unstructured grid with upscaling level of 35 (corresponding to 375 blocks)performs better than the uniform grid with upscaling level of 17 (corresponding to 786 blocks).Therefore, a much better performance can be achieved with a non-uniform unstructured gridof smaller size than a uniform grid, resulting in a computationally cheaper simulation.

The better performance observed for the unstructured (non-uniform) coarse grid com-pared to the uniform coarse grid is attributed to preservation of both critical heterogeneity

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Fig. 9 Water cut results on CSP 10th Model 2, layer 37 for fine grid, uniform, and non-uniform unstructuredcoarse grids for the two upscaling ratios of 17 and 35

Fig. 10 Oil recovery factor results on CSP 10th Model 2, layer 37 for fine grid, uniform, and non-uniform(unstructured) coarse grids for two upscaling ratios of 17 and 35

and flow areas, which are not necessarily preserved in the uniform coarse grid. In addition,use of unstructured grid and its flexibility led to better and more efficient local adaptation ofblock sizes.

6.2 Second Model: Synthetic Channelized Model

The second test model, taken from Chen et al. (2004), is one realization of a 2D synthetic chan-nelized system containing 100×100 gridblocks on Cartesian coordinate system. This modelwas generated using multi-point geostatistics (for the channels) and two-point geostatistics(for permeability distributions within each facies). This permeability model is characterized

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Fig. 11 Permeability map for synthetic channelized model

Table 3 Parameters used incoarse grid generation forsynthetic channelized model

Normalizedvorticity cut-off

dmin dmax ψmin ψmax

0.1 20 ft 70 ft 0.006 20

by high variations from 0.02 to 6.5×107 mD, a mean of 7×104 mD and isotropic normalizedcorrelation lengths varying from 0.05 to 0.5—Fig. 11. In this model, the correlated perme-abilities are not aligned with the Cartesian grid lines, but rotated almost at 45◦ with respectto the grid. This model represents a case in which the flow solution is highly affected by gridorientation and flow conditions in the finite difference scheme, as the preferred flow pathsare along high permeabilities, not the model coordinate system. High permeability variation,high permeability channel, and complex geometry of heterogeneities make it a challengingmodel to upscale using uniform grids and even non-uniform gridding to optimize gridblockboundaries for such a heterogeneous system.

The proposed coarse grid generation algorithm is applied on this model. Parameters usedfor generation of a coarse grid with upscaling level of 17 are listed in Table 3. The normal-ized single-phase vorticity map depicted in Fig. 12 highlights the critical areas to be refined.According to the map, the highest vorticity values are in the connected parts of the centralhigh permeability zones between injection and production wells where most of flow occurswhile in the rather homogeneous permeability regions, low flow regions and disconnectedhigh permeability areas they are small. Use of vorticity in this example is superior to velocityor flow as used in flow-based gridding because if in a region flow is low but permeability vari-ation is high, vorticity can be still high, and we require small gridblocks, while in a flow-basedtechnique, only high flow regions are resolved, and the rest of blocks are coarsened.

The vorticity mapped on permeability field is used to generate the background grid (seeFig. 13) for guiding the grid point insertion routine. The unstructured coarse grid generatedfor this model is shown in Fig. 14. Two-phase water/oil flow was simulated on this model aswell as on the fine grid and a uniform coarse grid with upscaling level of 17. Relative per-meability functions are the same as in the previous example. Water fractional flow (Fig. 15)and oil recovery (Fig. 16) predicted by these models show a nice match to fine grid results

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Fig. 12 Normalized vorticity map for synthetic channelized model

Fig. 13 Background grid generated for synthetic channelized model

for grid that is generated based on vorticity, emphasizing the success of the proposed methodin handling highly heterogeneous models. Time average oil production rate errors for theuniform and non-uniform unstructured coarse grids are 27% and 9.1%, and breakthroughtime error for these grids are 15.2% and 7.7%, respectively. Quantitatively, the non-uniformunstructured coarse grid generated by the method is preferable to uniform coarse grid.

6.2.1 Point Drive Injection/Production Scheme

All the gridding and flow simulation discussed above are based on line-drive scheme for bothinjection/production wells. If the injection/production wells are placed inside the model insingle gridblocks, re-gridding of the coarse grid model is expected to honor the new flowfield and to result in the generation of a different grid. For this purpose, synthetic modelis re-gridded based on point injection and production wells which are located at (200 ft,

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Fig. 14 Non-uniform unstructured Voronoi grid for synthetic channelized model

Fig. 15 Water cut results for synthetic channelized model for fine grid, uniform, and non-uniform unstructuredcoarse grids. Line-drive injection/production scheme

200 ft) and (800 ft, 800 ft), respectively, both on the lower branch of channel. Vorticity mapand background grid for the upscaling ratio of 35 are shown in Figs. 17 and 18. Figure 19exhibits the coarse grid generated with this configuration, which is clearly different from theone in Fig. 14. Two-phase flow simulation (Fig. 20) on the new non-uniform unstructuredcoarse grid, fine grid, and the uniformly coarsened model of the same number of gridblockscorroborates the success of the proposed method for grid generation. Quantitatively, the timeaverage oil production rate errors for the unstructured coarse grid and the uniform one are8.8% and 57%, respectively (Fig. 21).

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Fig. 16 Oil recovery factor results for synthetic channelized model for fine grid, uniform, and non-uniformunstructured coarse grids. Line-drive injection/production scheme

Fig. 17 Normalized vorticity map for synthetic channelized model. Point injection/production scheme

6.2.2 Effect of Different Types of Grid on Pressure Contours

For further investigation of the grid generated by vorticity-based coarse grid generation,single-phase flow pressure contours obtained from flow simulation in fine, non-uniformunstructured, and uniform grids with line injection for the channelized synthetic model areshown in Fig. 22. This simple quality check further corroborates that the grid generated bythe proposed algorithm based on vorticity succeeds in the identification of critical regionsfor refinement and well captures the heterogeneities. This check, before trying to model two-phase flow, is necessary to ensure that the coarse unstructured grid is satisfactorily producingfine grid single-phase pressure, and exhibits correct overall resistance to flow compared withfine grid.

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Fig. 18 Background grid generated for synthetic channelized model. Point injection/production scheme

Fig. 19 Non-uniform unstructured Voronoi grid for synthetic channelized model. Point injection/productionscheme

6.2.3 Effect of Well Location in Coarse Grid Generation

Change of the location of wells alters the orientation of streamlines and flow direction, whilepermeability system remains unchanged. Thus vorticity of single-phase flow, which is depen-dent on the magnitude and relative direction of permeability variation and velocity, will bedifferent. This difference is translated into different distributions of spacing parameter, andsubsequently, into generation of different coarse grids.

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Fig. 20 Water cut results for synthetic channelized model on fine grid, uniform, and non-uniform unstructuredcoarse grids. Point injection/production scheme

Fig. 21 Oil recovery factor results for synthetic channelized model on fine grid, uniform, and non-uniformunstructured coarse grids. Point injection/production scheme

In order to demonstrate this, we place the injection and production well in the first testcase at the bottom and top of the model, and seal the left and right boundaries. The algorithmis similarly applied to generate normalized vorticity map and background grid (Figs. 23,24). The difference between these figures and Figs. 5 and 6 is noticeable, and that is whythe generated coarse grid shown in Fig. 25 (with the same cut-off value for normalized vor-ticity) is dissimilar to the one in Fig. 8. This observation ensures that grid generation bythe proposed method is sensitive to the location of wells, as expected. In real applications,where the geological model is highly detailed, this issue becomes a disadvantage, though. Inthese cases, during the development of the field where, e.g., new infill wells are introduced,numerical grid has to be regenerated which means the re-computation of single-phase flowsolution on fine grid. Re-computation would be quite essential as well as advantageous for

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Fig. 22 Normalizedsingle-phase flow pressurecontours obtained from a finegrid b non-uniform unstructuredgrid, and c uniform grids forsynthetic channelized model

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Fig. 23 Normalized vorticity map of layer 37 from Model 2, SPE CSP, vertical injection/production

Fig. 24 Background grid generated of layer 37 from Model 2, SPE CSP, vertical injection/production:cut-off = 0.5

Fig. 25 Unstructured Voronoi grid of layer 37 from Model 2, SPE CSP, vertical injection/production

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Fig. 26 Effect of mobility ratio on the performance of unstructured coarse grid generated for synthetic chan-nelized model with line injection/production scheme. a Water cut, b Oil recovery factor

proper construction of coarse grid in any type of gridding which utilizes flow information.Since coarse grids constructed based on actual flow dynamics (as happens in flow-based orvorticity-based techniques) are superior to those constructed based on static description ofreservoir properties (as obtained from permeability-based techniques).

In fact single-phase flow simulation on fine grid is much less numerically intensive thantwo-phase flow simulation on the same grid; however, it can still be prohibitive for verylarge fine scale models in practice, e.g., tens or hundreds of million cells. The issue of re-computation can be alleviated, thanks to the recent advancements in streamline simulationtechnology in which multidimensional flow problem is decomposed into a 1D problem along

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Fig. 27 Effect of mobility ratio for synthetic channelized model with line injection/production scheme: timeaverage oil production error (ETAOPR, %) and breakthrough time error (EBTH, %) for vorticity-based coarsegrid

streamlines. This provides a much faster (still accurate) flow simulation than the conventionalfinite-difference, finite-volume methods.

Noting that a major step in the re-computation is the construction of background grid,there could be an alternative solution. Background grid can be constructed in coarse orfine grid. Based on our experience, structure of generated coarse grids from backgroundgrids obtained from original fine grid and a coarser model of fine grid are only slightlydifferent. In addition, usually single-phase flow is not as sensitive to block size as two-phase flow simulation is. These imply that instead of single-phase solution on fine grid,one can perform simulation on a slightly coarser grid. This can be a favorable practicalsolution for really large models (multi-million grid cells), to still be able to incorporateflow information into gridding procedure, while compromising a bit on fine grid solutionaccuracy.

6.3 Effect of Mobility Ratio on the Performance of Generated Coarse Grid

In order to assess the effect of two-phase flow properties on the robustness of coarse grid gen-erated and its process dependency, e.g., to change of injection fluid viscosity and properties,mobility ratio was varied, and two-phase flow was simulated for synthetic channelized model.This was performed for both line- and point-drive injection/production schemes. As shownin Figs. 26 and 27 (for first test model) and in Figs. 28 and 29 (for second test model), bychanging mobility ratio dramatically from 0.1 to 1000, the two-phase flow dynamic changessignificantly, e.g., fluid distribution and breakthrough time of displacing fluid, but this slightlyaffects the performance of the generated coarse grid. Coarse grid performance for high mobil-ity ratios, where fingering and oil bypass is more likely to occur, is as good as low mobility(favorable) displacements. For high mobility ratios, which are typically observed for viscousoil in real fields, this statement is more pronounced, as mobility ratio changes from 10 to

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Fig. 28 Effect of mobility ratio on the performance of non-uniform unstructured coarse grid generated forsynthetic channelized model with point injection/production scheme. a Water cut, b Oil recovery factor

1000 has insubstantial effect on the two-phase flow results. This observation could also beanticipated from vorticity analysis presented in “Vorticity Concept” section. Because for thishighly heterogeneous system, the chief controlling factor on vorticity is the heterogeneityterm used for coarse grid generation, and the grid structure remains robust even after alteringmobility of fluids.

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Fig. 29 Effect of mobility ratio for synthetic channelized model with point injection/production scheme: timeaverage oil production error (ETAOPR, %), and breakthrough time error (EBTH, %) for vorticity-based coarsegrid

7 Summary and Concluding Remarks

Vorticity measure calculated from single-phase flow was used to generate coarse non-uniformunstructured Voronoi grid with gridblocks of varying size with denser (finer) distribution athigh vorticity regions and coarser at low vorticity regions. Coarse grid distribution is deter-mined from upscaling ratio and by selecting a vorticity cut-off value. Gridblocks whose vor-ticity is above cut-off value are recognized as internal source elements from which size anddistribution of gridblocks are adjusted by use of advancing front method and, subsequently,by Delaunay triangulation techniques. The technique is applicable to general permeabilitysystems and employs an IMPES control volume finite difference numerical scheme to solveflow equations.

Layer 37 from SPE 10th CSP Model 2 as well as a synthetic channelized model wereused to evaluate the proposed method. Both models were highly heterogeneous and featuredwide permeability distribution and intersecting channels. Two-phase flow results on unstruc-tured coarse grid generated for both models showed a substantial improvement over simpleuniform grids and matched closely the fine scale behavior.

It was shown that relative direction of permeability variation and flow is of prime impor-tance, and the fact that well location strongly affects the coarse grid generation was demon-strated. The feature of vorticity is that it incorporates both permeability variation and flowvariables in coarse grid construction, so as to avoid unnecessary refinements at low flowregions as wells as avoiding to generate coarse gridblocks with high permeability variationswhich challenges upscaling of permeability.

The process dependency of generated coarse grids was also examined by changing mobil-ity ratio of the injected/produced fluids. It was shown that mobility ratio slightly impacts thequality of the grids generated, and re-gridding is not required if flow dynamics changes. Thisis of great importance and reduces cost of generating coarse grids.

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However, we have used a simple upscaling technique in the construction of coarse grids.Using a more accurate upscaling technique appropriate for actual flow, such as simulation-based, e.g., local flow-based, extended-local, or adaptive local-global techniques (Chen etal. 2003; Chen and Durlofsky 2006; Lambers and Gerritsen 2005), which introduce globalconnectivity into upscaling process, would further improve performance of the technique.

Finally, when applied to 3D systems and realistic field models, calculation of vorticityis straightforward and normally comes out of flow simulator using finite-difference/finite-volume techniques or streamline simulation. Background grid approach requires only minormodification for 3D extension and can be used to effectively distribute the grid points andsubsequently create Voronoi or other type of unstructured coarse grids. Nonetheless, Voronoigrid has its own limitations in 3D while other type of grids such as CVFE or its extensionare more suited for 3D applications. In order to address limitations of Voronoi grid in 3D,a layer-cake approach or 2.5D unstructured grid can be applied for vertical direction, whereVoronoi-type grids are generated only in 2D areal layers, by ensuring proper connection ofgrids vertically is attained. This can be a starting point to extend the technique to 3D systems.Noting that in 3D applications, vorticity vector has three non-zero components as opposedto 2D where only one non-zero component exists. In case of 3D flow, one should use boththe resultant vorticity magnitude and direction to guide the ADF approach. This should beproperly incorporated in the gridding algorithm.

In 3D, application of efficient upscaling would be even more critical than in 2D in viewof more flow complexities and more freedom of movement in 3D than in 2D. Nonetheless,use of more accurate upscaling techniques involves intensive computations, although theyare not yet readily available on unstructured grid.

Acknowledgements M. Evazi and H. Mahani gratefully acknowledge the partial support of this study bythe National Iranian Oil Company (NIOC).

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