Comput Geosci (2009) 13:187–214DOI 10.1007/s10596-008-9111-0

ORIGINAL PAPER

Upscaling of the permeability by multiscale wavelettransformations and simulation of multiphase flowsin heterogeneous porous media

M. Reza Rasaei · Muhammad Sahimi

Received: 20 April 2008 / Accepted: 15 September 2008 / Published online: 24 October 2008© The Author(s) 2008. This article is published with open access at Springerlink.com

Abstract We describe a new approach for simulation ofmultiphase flows through heterogeneous porous media,such as oil reservoirs. The method, which is based onthe wavelet transformation of the spatial distributionof the single-phase permeabilities, incorporates in theupscaled computational grid all the relevant data onthe permeability, porosity, and other important prop-erties of a porous medium at all the length scales. Theupscaling method generates a nonuniform computa-tional grid which preserves the resolved structure of thegeological model in the near-well zones as well as inthe high-permeability sectors and upscales the rest ofthe geological model. As such, the method is a multi-scale one that preserves all the important informationacross all the relevant length scales. Using a robustfront-detection method which eliminates the numericaldispersion by a high-order total variation diminishingmethod (suitable for the type of nonuniform upscaledgrid that we generate), we obtain highly accurate resultswith a greatly reduced computational cost. The speed-up in the computations is up to over three orders of

M. R. Rasaei · M. Sahimi (B)Mork Family Department of Chemical Engineering andMaterials Science, University of Southern California,Los Angeles, CA 90089-1211, USAe-mail: moe@iran.usc.edu

M. R. RasaeiDepartment of Chemical Engineering and Instituteof Petroleum Engineering, University of Tehran,Tehran 11365-4563, Irane-mail: mrasaei@ut.ac.ir

magnitude, depending on the degree of heterogeneityof the model. To demonstrate the accuracy and effi-ciency of our methods, five distinct models (includingone with fractures) of heterogeneous porous mediaare considered, and two-phase flows in the models arestudied, with and without the capillary pressure.

Keywords Two-phase flows · Geological model ·Upscaling · Wavelet transformations

1 Introduction

Computer simulation of multiphase flows in heteroge-neous and large-scale porous media, such as petroleumreservoirs, involves intensive computations (see, for ex-ample, [79]). Advances in measurement and estimationtechniques, together with considerable progress in thedevelopment of theoretical and computational methodsfor characterization of heterogeneous porous media,provide significant and accurate amount of data ontheir important properties. Based on such data, moderngeostatistical techniques [46] generate highly-resolvedgeological models of natural porous media. Due tocomputational limitations (mainly computation time),however, it is usually very difficult, if not impossible,to carry out simulation of multiphase flows using thegeological model. It is, therefore, necessary to upscalethe properties of the geological model’s grid blocks inorder to develop a coarsened grid that can be usedin the computer simulation with an affordable amountof computation time while yielding accurate results.Moreover, the correct appraisal of the potential pro-duction of an oil reservoir must take into account the

188 Comput Geosci (2009) 13:187–214

effect of the uncertainties in its properties. Productionforecast made by a reservoir’s stochastic model allowsone to quantify the effect of the uncertainties on itsperformance by simulating multiphase flows in manyrealizations of the reservoir—a very time-consumingtask. To reduce the simulation time, upscaling of therealizations of the reservoir’s geological model is, per-haps, the only feasible remedy.

To simulate the flow of a single fluid in a reservoir,the most important parameter is the spatial distributionof the absolute permeability. This aspect of the problemis now well understood (see, for example, [57, 74, 99]).Various classification of the upscaling methods is pos-sible based on the methods of their derivation (ana-lytical or numerical), the dependence on the boundaryconditions (local/extended-local, global/quasi-global,local/global), and the geological model employed (de-terministic versus stochastic).

Analytical methods have the advantage that they areusually faster than numerical algorithms that are basedon the numerical solution for the pressure distribution.They have the disadvantage that when applied to a sec-tor outside their strict domain of validity, they quicklybecome inaccurate. Numerical methods are, therefore,more accurate but may become computationally toointensive.

In the local approaches, the upscaled transmissibili-ties for the faces of the upscaled blocks are computedusing the generic flow solutions in the fine grid centeredaround the faces (see, e.g., [22, 69]. Such methods havebeen refined by several groups [40, 100]. The bound-ary conditions are typically of the Dirichlet type, orperiodic, imposed on the fine grid. Local methods mayproduce unsatisfactory results if large-scale connectedpaths exist within a reservoir. To improve the accu-racy, several approaches have been suggested, but theyare significantly more expensive computationally. Forexample, [101] derived coarse-scale properties basedon several global flows, while [41, 64] formulated theglobal upscaling as an optimization problem for a givenflow scenario.

The local/global method was recently suggested by[14] in order to improve the upscaled representationof large-scale connectivities in a given geocellular grid.In this method, a standard (extended) local methodis used to obtain the approximate upscaled perme-ability or transmissibility field. Then, a coarse globalsimulation is carried out. When the solution is inter-polated for the local scales, the appropriate Dirichletboundary condition is obtained for the next iteration.The iteration between the global and local scales iscontinued until the coarse field is converged. Chen

and Durlofsky [13] developed an adaptive local/globalupscaling procedure that provides coarse-scale param-eters that are adapted for more general types ofglobal flows (such as, for example, those driven bya well, or a particular set of boundary conditions).Gerritsen et al. [34] introduced a multilevel local/globalmethod, which combines the local/global upscaling inorder to improve the representation of the large-scaleconnectivities.

Simulation of multiphase flows, however, also in-volves adjustments to the flow of the various fluidphases through the connected blocks of the upscaledgrid. To accomplish this task, many methods havebeen proposed, including upscaling techniques thatemploy the so-called pseudo-functions, the dual-meshapproach, and coarsening methods that generate a com-putational grid with nonuniform blocks. The pseudo-function technique is perhaps the most widely usedmethod of upscaling of multiphase flows. It consistsof replacing the original saturation-dependent relativepermeabilities at the length scale over which the dataare measured by fictitious functions that mimic the fluidflow process in the upscaled computational grid. Vari-ous versions and refinements of this basic concept havebeen proposed [19, 37, 43, 90]. Emmanuel and Cook[31] proposed pseudo-functions for well completionsin an upscaled grid. The most widely used method ofcalculating the dynamic pseudo-functions was proposedby [49] who also introduced pseudo-capillary pressurefunctions. Despite its popularity, the KB method doesnot yield accurate results for strongly heterogeneousreservoirs; [48, 51] presented methodologies suitablefor heterogeneous reservoirs that contain a hierarchyof relevant and distinct length scales. Stone [89] andothers [9, 39] used the average total mobility in or-der to avoid computing the phase potentials in theupscaled grid that the KB method requires. Pickupand Sorbie [70] showed that one must characterize aheterogeneous reservoir by a tensorial effective relativepermeability and, hence, tensorial pseudo-functions. Areview of such concepts and techniques was given by [8](see also [7]).

An accurate alternative to pseudo-functions, knownas the dual-mesh (DM) method [33, 95], solves for thepressure distribution in the upscaled grid and includesthe fine-scale heterogeneities in the computation ofthe phase saturations. Arbogast and Bryant [4] intro-duced a version of the DM method that upscales thetransmissivities. They used a mixed finite-element (FE)method in which the fine-scale effects are localized bya boundary condition at the boundaries of the coarseblocks. The influence of the small-scale heterogeneities

Comput Geosci (2009) 13:187–214 189

is then coupled with the coarse-scale effects using theappropriate Green functions. Audigane and Blunt [6]presented a 3D extension of the method using a finite-difference (FD) technique. The DM methods make noassumption regarding the relative significance of theviscous, capillary and gravitational forces, and gener-ate negligible numerical dispersion, but their computa-tional speed-up over the original geological model ishardly larger than 10. Hou and Wu [42] employed aFE approach and constructed specific basis functionsthat capture the influence of the heterogeneities atsmall scales. The method was further refined by [15].Lee et al. [52] developed a flux-continuous schemefor 2D models and extended the method [53] to 3Dmodels. Further improvement of the method was sug-gested by [44]. More recently, [45, 56] proposed amultiscale finite-volume approach for computing theeffective coarse-scale transmissibilities with tensorialpermeabilities for 3D unstructured grids.

Generally speaking, although one can accuratelycompute the effective absolute permeabilities of theupscaled grid blocks by uniform coarsening, one alsoobtains an excessively smooth upscaled permeabilitydistribution that does not honor the important ef-fect of the distribution’s tails at the fine scale, hence,leading to inaccurate predictions for such importantquantities as the breakthrough time of the displacingfluid. Durlofsky et al. [23] introduced an upscalingmethod whereby a grid of finer resolution is used inthe regions of high fluid velocities, but upscaled, ho-mogenized description is utilized for the rest of theflow domain. Similar to the DM method, in Durlofskyet al.’s approach, the relative permeabilities are notupscaled. The fine-scale relative permeability functionsare used for the upscaled grid blocks, hence, making thetechnique process-independent, a procedure consistentwith the homogenization theory of multiphase flowsat large scales [3, 10, 77]. Nonuniform grid coarseningoffers significant computational speed-up and generalapplicability but at the cost of increasing numericaldispersion and decreasing accuracy for non-separablelength scales.

Numerous grid selection methods rely on dynamicresponses in order to identify the grid blocks in thegeological model through which the fluids pass [12, 23,96, 97]. But, they are effective for accurately preservingthe essential features of fluid flow at the fine scaleonly when the boundary conditions remain relativelyconstant. Because the pressure field in the near-wellzones usually changes severely in the radial direction,such upscaling approaches are not usually accurate ifthey are used both in such zones and in the regions far

from the wells. To address this problem, [24] developedan approach to calculate the transmissivities and wellindices for single-phase flows based on the solution tothe local well-driven flow.

We may also distinguish upscaling methods in termsof the stencils used to compute the flux across a block’sface. In the two-point flux approximation (TPFA), theflux is computed based on the pressures in the twoblocks that share the same face. In the more accuratemultipoint flux approximation (MPFA), on the otherhand, additional blocks are involved, which lead natu-rally to larger stencils than those in the TPFA methods.In 3D structured grids, for example, the MPFA methoddeveloped by [1] results in a 27-point stencil for the cell-centered pressure equations, compared with the seven-point stencils used in the TPFA methods. The approachhas been studied by others as well [30, 53].

Aside from their high computational costs, theMPFA methods, when utilized in cases with strong per-meability anisotropy, suffer from nonmonoticity. Thatis, the solution for the pressure distribution may containunphysical oscillations [65]. The TPFA methods alsosuffer from a somewhat similar shortcoming in that theyyield inconsistent approximate solutions to the single-phase pressure distribution when utilized in the samecases. Perhaps surprisingly though, the TPFA methodscombined with the local/global procedures have beenshown [13, 14] to provide accurate upscaled transmis-sibilities, when utilized in modeling flow in reservoirswith complex geologies. On the other hand, [50] pre-sented a compact multipoint method that allows theMPFA stencil to vary spatially.

In this paper, we develop a new and highly efficientapproach for simulation of multiphase flows based onthe use of wavelet transformations. The method is amultiscale approach that acts as an “intelligent” gridgenerator at all the relevant length scales that are in-corporated in the geological model. For the cases thatwe study, the method carries out the upscaling solelybased on the spatial distribution of the single-phasepermeabilities and yields very accurate results. Theplan of this paper is as follows: in Section 2, we brieflydescribe important properties of wavelet transforma-tions that we will use in our computations. Section 3describes upscaling of the geological model based onwavelet transformation of the spatial distribution ofthe permeabilities, while in Section 4, we describe thegeological model that we use in our work. The nu-merical technique that we develop and utilize to solvethe governing flow equations is described in Section 5,after which the results are presented and discussedin Section 6. Section 7 discusses the speed-up in the

190 Comput Geosci (2009) 13:187–214

computations that the wavelet-based method yields.The paper is then concluded with a discussion of theextension of the method to more complex models ofporous media.

2 Wavelet transformations

We assume that the reservoir’s heterogeneity is rep-resented by the distribution f [K(x)], where K(x) isthe single-phase permeability of a grid block of thereservoir’s geological model with its center at x; f (K)

contains correlations at all the length scales that areincorporated in the geological model. Fourier trans-formation (FT) is inadequate for analyzing such databecause all the important information about the data’slocalization properties will be lost in such an analy-sis. It is essential to represent such data in terms oflocalized functions. The requirement of locality was aprime motivation for developing the wavelet analysisand wavelet transformations (WT) since (see below)wavelet functions are nonzero over only limited inter-vals and, therefore, are localized.

In general, one must distinguish a continuous WT(CWT) from a discrete WT (DWT). A CWT is (similarto a continuous FT) used for analyzing and detect-ing unusual features of a data set. A DWT, on theother hand, is (similar to a discrete FT) used for datacompression and reconstruction. A WT utilizes a func-tion ψab (X), obtained by translating and rescaling of amother wavelet ψ(x), which is selected from a family ofsuch functions:

ψab (X) = 1

ad/2ψ

[(x − b)

/a], (1)

where a > 0 is the rescaling parameter, b representstranslation of the wavelet, and d is the system’s di-mensionality. As the word wavelet suggests, ψ(x) hasa wavelike behavior within a limited domain. Moregenerally, each of the coordinates x = (x, y, z) may haveits own distinct rescaling parameter, a = (ax, ay, az),in which case ad/2 is replaced by √

axayaz. The choiceof a−d/2 in Eq. 1 is not essential (particularly for theproblem studied in this paper). In fact, a normalizationfactor a−1 is often used in 1D which has the advantageof giving more weight to the small scales. A wavelet isusually required to have a zero mean:

∫ ∞

−∞ψ (x) dx = 0 (2)

More generally, ψ(x) is often required to have a num-ber of vanishing moments:∫ ∞

−∞

∫ ∞

−∞

∫ ∞

−∞xα yβzγ ψ (x, y, z) dxdydz = 0,

0 ≤ α + β + γ ≤ M (3)

which improves the efficiency of the wavelet at detect-ing singularities or unusual features of a dataset. Thepossibility of selecting the wavelet from a family ofsuch functions affords one great flexibility in choosinga suitable wavelet for analyzing a given dataset.

The WT of K(x), also called its wavelet detail coeffi-cient, is then defined by

D (a, b) =∫ ∞

−∞K (x) ψab (x) dx

= 1

ad/2

∫ ∞

−∞K (x) ψ

[(x − b)

/a]

dx (4)

Equation 4 exhibits a fundamental property of awavelet analysis: by computing the WT of K(x), us-ing a shifted and rescaled wavelet, one analyzes K(x)at increasingly coarser (a > 1) or finer (a < 1) lengthscales. Therefore, by devising a proper scheme basedon the DWT of K(x), one can, at any length scale ofinterest, systematically compress information on K(x)in those parts of the computational grid (representingthe geological model) where detailed description is notneeded and obtain a new grid with larger blocks. Theprecise manner by which the compression is done is de-scribed below. The transformation K(x)→ D(a,b) maybe inverted exactly, yielding a reconstruction formula(which is important to the problem that we consider inthis paper; see below):

K (x) = C−1ψ

∫db

∫1

ad+1ψab (x) D (a, b) da, (5)

in which Cψ is a normalization constant, and the do-mains of the integrals represent all possible values of aand b. Equation 5 implies that a WT decomposes a dataset as a linear superposition of the wavelets ψab withcoefficients D(a,b), and that the natural measure in theparameter space (a,b) is dbda/ad+1 which is invariantnot only under scale translation but also under dilation.

Since in the upscaling problem, we utilize a com-putational grid, a DWT is used which is obtained byrestricting the parameters a and b to the grid points.Hence, in 1D for example, one writes,

D j,k = 2− j/2∫

ψ(2− jx − k

)K (x) dx, (6)

where j and k are integers. One can construct functionsψ for which the set {ψ j,k, j, k} is an orthonormal basis

Comput Geosci (2009) 13:187–214 191

of in the real space. Such a basis has all the importantand useful properties of the wavelets, including spacelocalization, and yields fast computational algorithms.The construction of the basis uses the fact that almostall the wavelets’ orthonormal bases are derived from amultiresolution analysis, which we now describe briefly.

D j,k1,k2 contains information only about the differ-ence or contrast between two approximations of thesame function at two successive length scales. Themost accurate approximation of a function at a fixedscale is given by another function called the scalingfunction φ(x) such that, for example in 1D, the set{φ j,k ≡ φ

(2 jx − k

)}is orthonormal, with k being an

integer and the definition of φ j,k(x) being similar to thatof ψ j,k(x). Whereas the mean value of ψ(x) over theentire space is zero, the mean value of φ(x) is unity overthe same space. In analogy with D j,k1,k2, the waveletapproximate or scale coefficients are defined by

S j,k =∫ ∞

−∞φ j,k (x) K (x) dx, (7)

As described below, the scale and wavelet functions areinterrelated.

Consider, first, the 1D wavelets. The orthonormalwavelets ψ are constructed based on the two-scale (alsoknown as the scaling or refining) equation:

φ (x) = √2

∞∑

n=−∞hnφ (2x − n) , h = 〈φ1, n| φ〉 (8)

from which the orthonornal wavelets are generated by

ψ (x) = √2

∞∑

n=−∞(−1)n−1 h−n−1φ (2x − n) , (9)

where the hn are usually called the filter coefficients.One important family of orthonormal wavelets withcompact support, i.e., those that are nonzero over onlysmall intervals of x, is the Daubechies set of wavelets oforder M [20, 21] (usually referred to as DBM), used inthis paper. They possess the property that their first Mmoments are zero. The scaling function φ(x) is relatedto those at the finer length scales by an equation similarto Eq. 8:

φ (x) = √2

L−1∑

n=0

hnφ (2x − n) , (10)

where L = 2M and, similar to Eq. 9,

ψ (x) = √2

L−1∑

n=0

mnφ (2x − n) . (11)

The quantities mn, which are also filter coefficients,are related to hn by (compare Eqs. 9 and 11) mn =(−1)n hL−n−1, with n = 0, 1, . . . ,L−1. They are usuallynonzero for only a few values of n since the waveletshave compact support.

The simplest wavelet is the Haar or DB1 waveletwhich, in 1D, is defined by

ψ0,1 (x) =⎧⎨

⎩

1 0 ≤ x < 1/

2−1 1

/2 ≤ x < 1

0 otherwise.

(12)

The filter coefficients of the Haar wavelet are (h0, h1) =(1/√

2)(1, 1) and, therefore, (m0, m1)=

(1/√

2)(−1, 1)=

(−h1, h0). The factor 1/√

2 (often not included) is toensure orthonormality between φ and ψ . Press et al.[71] provided an extensive list of the filter coefficientsfor a variety of wavelets.

Using the Haar wavelet, we demonstrate a key prop-erty of WTs. It is not difficult to see that the Haarwavelet expresses the approximation to a function f bywavelets by replacing an adjacent pair of steps by onewider step and one wavelet. The wider step measuresthe average of the initial pair of steps, while the wavelet,formed by two alternating steps, measures the differ-ence between the initial pair of steps. Thus, the sumof two adjacent steps with one half width produces thebasic unit step function, φ0,1 = φ0,1/2 + φ1/2,1. Similarly,the difference of the two steps yields the correspondingHaar wavelet, ψ0,1 = φ0,1/2 − φ1/2,1. The decompositionis a fundamental property of a WT: a WT decom-poses a function K(x) into two separate componentsor functions—a scaling function φ(x) and a waveletψ(x)—both acting on K(x). The two functions separatethe components into averages and differences, with a“wavelength” equal to the “window” over which φ andψ are nonzero. Moreover, a WT is recursive so thatit can be applied in succession to any set of averagesproduced using the wavelets to generate other levels ofaverages and details, and so on. The averages computedby a WT are not, in general, the arithmetic averages butdepend on the wavelet function used. Only the Haarwavelet produces the arithmetic average. These prop-erties are the key to the application of a multiresolutionWT to the problem studied in this paper.

Two-dimensional wavelets are usually built in sep-arable form by tensor products of the 1D wavelets(sometimes referred to as the Cohen–Daubechieswavelets). Therefore, in 2D, one has one scalingfunction:

φ j,k1,k2 (x, y) = φ j,k1 (x) φ j,k2 (y) , (13)

192 Comput Geosci (2009) 13:187–214

and three wavelet functions:

ψ(1)

j,k1,k2(x, y) = φ j,k1 (x) ψ j,k2 (y) , (14)

ψ(2)

j,k1,k2(x, y) = ψ j,k1 (x) φ j,k2 (y) , (15)

ψ(3)

j,k1,k2(x, y) = ψ j,k1 (x) ψ j,k2 (y) , (16)

where the superscripts 1, 2, and 3 indicate, respec-tively, the correspondence of the wavelets with thehorizontal, vertical, and diagonal changes in the data.The extension to 3D wavelets is straightforward: Onehas seven detail and one scale coefficients. All theresults reported in this paper were obtained using theDaubechies-four (DB4) wavelet. The choice was madebased on the efficiency of the computations and accu-racy of the results. Using higher-order wavelet does notonly improve the accuracy but also increases the com-putations’ time. The filter coefficients (h0, h1, h2, h3)

of the DB4 wavelets are 18

[(√2 + √

6),(3√

2 + √6),(

3√

2 − √6),(√

2 − √3)]

.

3 Wavelet scale up of the single-phase permeabilities

One major goal of this paper is to show that in manycases, the wavelet-based upscaling that uses only single-phase flow properties is accurate and efficient for manymodels of highly heterogeneous reservoirs. Therefore,we upscale only the single-phase permeabilities of thegrid blocks in the geological model.

While, undoubtedly, there may be cases for whichsome multiphase flow properties might need to be up-scaled, for the cases that we study in the present paper,as well as the three-dimensional model reservoir thatwe have studied elsewhere [73], namely, the SPE-10model [16], there seem to be no need for upscaling ofany multiphase properties.

The wavelet upscaling of the single-phase permeabil-ities was developed in our previous papers [26, 60, 82]which, for completeness, is briefly described here. Weconsider 2D models of oil reservoirs (or other typesof porous media); its extension to 3D which poses noconceptual difficulty over what we describe here. Themethod has been applied elsewhere [73] to upscaling ofthe SPE-10 model, geological model of an oil reservoirin which the single-phase permeabilities vary over eightorders of magnitude. The geological model is repre-sented by a square grid of equal-size blocks, which maybe obtained by discretizing the governing equations bythe standard five-point FD approximation or from afinite-element discretization of the governing equations

[60]. The dimensions of the grid blocks depend onthe resolution of the distribution f (K). Each squareblock is assigned a permeability K selected from thedistribution f [K(x)] (the extension to the case in whicheach block is characterized by a permeability tensoris straightforward; see below), taking into account thecorrelation between the blocks’ permeabilities.

To begin the upscaling, we first compute the DWTof K(x). Associated with the DWT at every block withits center at x = (k1, k2) [at x = (k1, k2, k3) in 3D] arefour wavelet coefficients (eight in 3D). Following Eqs. 6and 7, the four wavelet coefficients are, in more preciseforms, given by

Sl,k1,k2 =∫

�

K (x) φl,k1,k2 (x) dx, (17)

D(�)

l,k1,k2=

∫

�

K (x) ψ(�)

l,k1,k2(x) dx, (18)

where l is the level of upscaling (l = 0 represents the ge-ological model), and � is the domain of the problem. Asmentioned above, Sl,k1,k2 contains information aboutthe (weighted) average permeability at x at a fixed scaleof the grid, while the detail coefficients D(�)

l,k1,k2measure

the contrast or difference between K(x) of a block withits center at x in the coarser scale and those of itsneighbors in the immediate finer scale (one with blockshalf the size of the coarser scale), with � = 1, 2, and 3 (in3D, � = 1, L, 7) measuring the contrasts in K betweenthe blocks in the x, y, and the diagonal directions,respectively. Thus, a large Sl,k1,k2 indicates that the per-meability at x = (k1, k2) is large, while large values ofthe detail coefficients indicate large contrasts betweenthe permeabilities of two neighboring blocks in twosuccessive scales. Accurate and efficient computationsthen require that we identify such zones of the grid, andupscale the rest of the grid, as their representation atthe coarser scale suffices for taking into account theirinfluence on the fluid flow process.

To make the identification, two thresholds, εs and εd,are introduced [26, 60] and are set as a fraction of theircorresponding largest values throughout the grid. Thescale coefficient of each block is then examined. If it islarger than εs (implying that the block’s permeabilityis large), it is left intact, and the next grid block isexamined. If, however, the examined scale coefficientis smaller than εs, the associated detail coefficients areexamined and set to zero if they are smaller than theirthreshold εd. Setting D(�)

l,k1,k2means that the neighbor

of the block centered at (k1, k2) and in the direction�, which in the finer-scale grid is only one block (orone diagonal block) away from it, is not needed and,

Comput Geosci (2009) 13:187–214 193

therefore, the two blocks merge to form a block twiceas large. Sweeping the grid once and examining allthe scale and detail coefficients result in the merger ofmany grid blocks, hence, generating a new grid withfewer blocks. The blocks’ sizes in the new grid are nolonger equal, as some of the blocks in the fine-scale gridhave merged, while others have remained intact.

As mentioned earlier, one important property of aWT is its recursive nature. That is, one can repeatedlyapply the transformation to a partially compressed setof data to obtain another more compressed set. Takingadvantage of this property, the new grid is upscaledagain by applying the DWT to the scale coefficients,computed at the previous step, which contain informa-tion about the permeabilities in the current (partially)upscaled grid, and calculating a new set of four coef-ficients for each block of the (partially) upscaled grid.The grid is then swept again to merge some of theblocks and form larger blocks. Typically, after a fewsweeps (levels of upscaling l) the grid can no longerbe coarsened efficiently. In practice, the number ofupscaling levels depend on the broadness and structure(with or without correlation) of f (K).

It also depends on the nature of the problem. Thereduction in the number of the blocks depends on thethresholds εs and εd. Clearly, the higher the two thresh-olds, the larger is the number of the grid blocks that aremerged to form larger upscaled blocks and, hence, thesmaller is the number of grid points that are utilizedin the flow simulations. Instead of the permeabilityK(x), a WT can be applied to any other geologicalor petrophysical property that one may wish to basethe upscaling on. For example, one may upscale thegrid based on the distribution of the local flow veloci-ties in single-phase flow through the model. Moreover,the structure of the grid around the (injection and/orproduction) wells is important. Thus, we may considertwo upscaling scenarios: in one, the geological modelis upscaled based only on the wells’ positions, implyingthat the blocks are upscaled uniformly in the regionsfar from wells, and the fine grid structure in the near-well regions is maintained (or coarsened less). In thesecond scenario, we consider both the wells’ positionsand the permeabilities in the upscaling process. Someof the upscaling methods developed in the past alsoconsider the position of the wells in the grid.

After generating the upscaled blocks, an importantissue is computing their effective permeabilities, whichcan be done by many methods. One method is based onthe analogy between fluid flow and electrical currents.For example, a configuration that consists of four neigh-boring square blocks that make a larger square block,each having its own permeability, is replaced, after

upscaling by a larger block with an effective permeabil-ity K′ given by the following general form [67]:

K′ = K′ (Ki, Kij, hi, h j)

(19)

where Ki is the permeability of block i with, i = 1–4,Kij = Ki + K j, and hi and h j are the linear sizes of theblocks i and j. The appearance of the blocks’ linearsizes hi and h j is necessitated by the rescaling of thesmaller blocks into a larger block. Since the first levelof upscaling involves the equal-size blocks of the initialfine grid, all the linear sizes hi and h j are equal andcancel out. But, the upscaling beyond the first levelinvolves unequal-size grid blocks, which necessitatesintroduction of hi and h j.

If the permeabilities are direction-dependent, thenthe transformation (Eq. 19) is used once for each di-rection in order to compute the equivalent direction-dependent permeabilities. The second method is basedon reconstruction of f (K), i.e., computing its inverseWT after some of the scale and detail coefficients havebeen set to zero. The effective permeabilities of the up-scaled blocks are then assigned based on reconstructedf (K). We used both methods and found that the differ-ence between the two is negligible.

In general, starting with the geological model, thefinal upscaled grid will contain a distribution of blocksizes, as a large number of the blocks in the geologicalmodel merge at various levels of upscaling to formlarger blocks. Then, blocks of unequal sizes may be-come neighbors (see below). Such blocks (and the gridpoints that represent them) have only three (or fewer)neighbors (in 2D); therefore, the governing equationsfor fluid flow cannot, at such points, be discretizedusing the standard FD approximation. To overcomethis difficulty, we use the finite-volume method which isbest suited for use in computational grids that containunequal blocks [38, 82]. We will come back to this pointshortly.

Applications of the WT to problems related to petro-leum reservoirs and other types of porous media haverecently been increasing. In addition to analyzing theseismic data which were one of the prime motivationsfor developing the WT, the applications also includenumerical simulation of flow and transport equations[62], upscaling of a conductivity or permeability field[26, 60, 82], pressure transient analysis [5, 25, 28, 47, 87],upscaling of miscible displacements in heterogeneousporous media [27], reservoir characterization [11, 55,59, 63, 66], identification of the fracture density alongwells in oil and gas reservoirs [81], as well as problemsinvolving turbulent transport and chemical reactions inthe atmosphere [38]; for a recent review, see [78, 80].

194 Comput Geosci (2009) 13:187–214

We point out that [17] also attempted to use the WTfor upscaling using a method different from what wedescribe here. In addition, in their method, the grid wasupscaled uniformly throughout (that is, the upscaledblocks all have the same size). As such, the method isnot much different from many of the previous upscalingmethods that did not use the WT.

4 The geological model

To put the method under a stringent test, we use asynthetic geological model in which not only is thedistribution f (K) of the local permeabilities relativelybroad (typically distributed over three orders of magni-tude) but also contains long-range correlations. To thisend, we generate the permeabilities by the fractionalBrownian motion (FBM), a stochastic process that in-duces long-range correlations in the permeabilities. Theextent of the correlations that are generated by theFBM is as large as the system’s linear size. Briefly,the FBM is characterized by a spectral density given by

S (ω) = a (d )(∑

i ω2i

)α , (20)

where a(d) is d-dependent constant, and α = H + d/2,with d being the dimensionality of the system. TheHurst exponent H is such that H > 1/2 (<1/2) impliespositive (negative) correlations, while for H = 1/2 thesuccessive increments in the FBM-generated values areuncorrelated. There is considerable evidence that theFBM-type correlations are ubiquitous in oil reservoirsand groundwater aquifers (see, for example, [61]). Togenerate a permeability distribution for a stratifiedreservoir, we rewrite the 2D power spectrum of theFBM as,

S (ω) = a (d )(

a1ω2x + ω2

y

)α , (21)

where a1 is a numerical constant such that witha1 = 1, one obtains stratifications that are more or lessparallel to the x-direction. Extension of Eq. 21 to 3Dis straightforward. We emphasize that the upscalingmethod described in this paper is independent of thepermeability distribution f (K).

To provide a simple example, consider a 1D systemwith 16 blocks of different permeabilities generated bya 1D FBM. The synthetic sequence of the K values(level l = 0) is given by, (0.60, 0.916, 0.845, 0.60, 0.605,0.996, 1.0, 0.91, 0.83, 0.463, 0.29, 0.054, 0.048, 0.217,0.0628, 0.0257), so that the largest and smallest K differ

by a factor of about 40. After one level of coarsening(l = 1), using εs = εd = 0.9 one obtains,

(0.758, 0.727, 0.605, 0.996, 1.00, 0.91, 0.83, 0.463, 0.17,

0.13, 0.044),

hence, reducing the system to an eleven-block system.The K values for the coarsened blocks were calculatedby reconstruction, i.e., by computing the inverse DWTof the distribution of K, after setting to zero the ap-propriate wavelet scale and detail coefficients. Note, forexample, that the first two blocks on the left side (withK = 0.60 and 0.916) merged, as did the third and fourthblocks, while blocks number 5–10 remained intact. Thesecond level of upscaling (l = 2) yields,

(0.742, 0.605, 0.996, 1.00, 0.91, 0.83, 0.463, 0.17, 0.088),

which is a nine-block system. The upscaling can con-tinue further until it reduces the system to 2–3 blocks.

It might seem from the foregoing simple examplethat a direct upscaling algorithm based on the blocks’permeabilities and their contrast with those of theirneighbors may result in the same coarsened grid. How-ever, that is not the case. It is only in the wavelet spacethat the differences between the blocks’ permeabilities,and their absolute values can be used as a criterionfor upscaling. This is because, as is well-known in thetheory of WTs, a function may have a small value ata given point but have large wavelet scale and detailcoefficients in the wavelet space and vice versa. Thisis similar to image processing with the wavelets whichis done by digitizing the image, computing its WT,and compressing it by a method similar to what wedescribed above: the compression is not done in thereal space but in the wavelet space. Otherwise, thecompressed image will be distorted.

5 Numerical simulation of two-phase flows

We simulate a water-flooding process as an exampleof a typical two-phase flow in a heterogeneous porousmedium, such as an oil reservoir, and solve the problemin both the geological and upscaled models in order toassess the accuracy of the upscaling method. Combiningthe Darcy’s law and the mass conservation equationand ignoring the effect of gravity yield the followingequations for two-phase flow of oil and water in areservoir:

B−1o ∇. (Kλo∇�o) − qo

ρo= ∂

∂t

(ϕSo

Bo

), (22)

B−1w ∇. (Kλw∇�w) − qw

ρw= ∂

∂t

(ϕSw

Bw

), (23)

Comput Geosci (2009) 13:187–214 195

where, �w = Po − Pc, with Pc = Po − Pw, Po and Pw

being the pressures in the oil and water phases, λo andλw are, respectively, the mobility of the oil and water,and (Bo,qo) and (Bw, qw) are their corresponding for-mation volume factor and flow rate, respectively. Thedensities ρo and ρw are evaluated under the standardconditions. We then combine Eqs. 22 and 23 to obtaina single equation for Po, containing no explicit timederivatives of the phase saturations:

Bo

[∇.

(K

λo

Bo∇ Po

)− qo

ρo

]

+ Bw

[∇.

(K

λw

Bw(∇ Po − ∇ Pc)

)− qw

ρw

]= ϕC

∂ Po

∂t

(24)

Where Ci = −B−1i ∂ Bi

/∂ Pi(i=o, w) represents the

phase compressibility, Cr = −ϕ−1i ∂ϕ

/∂ Po is the rock’s

compressibility, and C = Cr + CoSo + CwSw.The governing equations for Po and So are solved

by a finite-volume (FV) method, which is most suitablefor the type of computational grid that we generate bythe upscaling method. Although most of the currentcommercial simulators for reservoir simulation utilizeFD approximation to the governing flow equations, dueto its great flexibility in handling a computational gridwith unequal blocks, use of the FV method is alsobecoming increasingly popular. In fact, the commer-cial simulator FLUENT (used for computational fluiddynamics) utilizes the FV method.

To utilize the FV method, we first integrate bothsides of Eq. 24 over a control surface A (control volumein 3D), to obtain∫ ∫

Cϕ∂ Po

∂tdA

=∫ {

Bo

[∇.

(K

λo

Bo∇ Po

)− qo

ρo

]

+Bw

[∇.

(K

λw

Bw(∇ Po − ∇ Pc)

)− qw

ρw

]}dA

(25)

The left side of Eq. 25 is evaluated by assuming thatthe oil pressure in any block can be represented by anaverage value. Likewise, the flow rates qo and qw are as-sumed to represent the average rates in any gridblock.The time derivative is then approximated by a suitableFD form. Therefore, the left side of Eq. 25 is written,for example, as, Cϕ�S

(Pn+1

i − Pni

)/�t, if we use a for-

ward FD approximation, where �S is the surface of theblock, Pn

i is the pressure after n time steps at grid point irepresenting a block, and �t is the time step. One, then,

invokes the divergence theorem to convert the surfaceintegrals on the right side of Eq. 25 to line integrals.Since the upscaled grid contains blocks of various sizes,all the terms on the right side of Eq. 25 after theirconversion to line integrals via the divergence theoremmust be evaluated carefully. Consider the configurationof the blocks shown in Fig. 1. For the western side of thebase block b we write the line integral as,

∫

wK

λo

Bo∇ Pody =

N∑

j=1

(� j + �b

� j/K j + �b /Kb

)(λo,up

Bo

)m

wn

×(

P j − Po

� j + �b

)Sb j (26)

Here, Sb j is the common surface (side) between twoadjacent blocks j and b , � j and �b are, respectively,the cell-to-face distances for the neighbor cell j and thebase cell b ; N ≥ 1 is the number of neighboring blockson the western side of the base cell b (N = 1 if the twoblocks have the same size), (P j, K j) and (Pb , Kb ) arethe average pressures and permeabilities representingblocks j and b , and λo,up is the mobility of the oil phasein the front block. The superscript m is the time stepnumber at which the quantities are evaluated.

The discretization represented by Eq. 26 results in alocal leading truncation error on the order of O(1/h),when applied to h̃ adaptive nonuniform grids [32, 72].The analysis of [29] indicated that in grids with largeaspect ratios the truncation error depends on the gridinterface and aspect ratios, the permeabilities, and thepressure gradient acting tangentially on the grid inter-face (between two unequal blocks). Thus, in order toeliminate such errors (or reduce them to very smallvalues), we followed the approach suggested by [29].Consider, for example, the configuration of the blocks

j

b

lb

lj

j

j

lb

lb

lj

lj

Fig. 1 A base block b and its “western” neighbors j in the finite-volume formulation

196 Comput Geosci (2009) 13:187–214

a

c

b

Fig. 2 Three blocks of unequal sizes used for computing thefluxes between them (see Eqs. 27–29)

shown in Fig. 2. To calculate the flux between a and c inthe figure, we use the standard expression based on theharmonic averaging of the blocks’ permeabilities:

Jac = 2hx

hy

Ka Kc

Ka + Kc(Pa − Pc) , (27)

whereas to compute, for example, the flux between aand b we use,

Jac = 4hx

hy

Ka Kb Kc

4Ka Kb +Ka Kb +Kb Kc

(Pb −(Pa+Pc)

/2),

(28)

with a similar expression for Jab , where hx and hy arethe blocks’ linear dimensions at the finer scale. In anupscaling method which is not based on the waveletfunction, Eq. 28 can be problematic. For example, ifthe permeability of either block a or c is very small,then the flux across both faces ab and cb will be verysmall or zero. More generally, if the faces betweenblock b , block a, and block c are of equal size, thenthe flux out of block b is evenly distributed betweenthe faces, which may lead to nonphysical oscillationsin the pressure field. In the absence of the waveletupscaling, an accurate way of addressing the issue is bythe use of the so-called L method [2].

However, the wavelet-based method proposed hereautomatically takes into account such effects. Suddenchanges in the blocks’ sizes and properties are correctlyaddressed by using the DB4 (or higher-order) waveletfunctions, which consider extra blocks around the cen-tral block under examination for upscaling. The suddenchanges will be reflected in the corresponding waveletscale and detail coefficients, which may then lead to noupscaling at all of the central blocks and its merger withits neighbors.

In any event, Eq. 26 is used only when equal-sizeblocks are neighbors (N = 1), but for the case in

which unequal blocks (with an interface ratio of 2) areneighbors, Eq. 26 is modified to∫

wK

λo

Bo∇ PodS

=N∑

j=1

4

(Ka Kb Kc

4Ka Kb + Ka Kb + Kb Kc

)

×(

λo,up

Bo

)(Sb j

lb

)(Pb − (Pa + Pc)

/2), (29)

As described by [29], the extension to grids of largerinterface ratios is straightforward.

The simulator first solves the governing equationsfor the pressure Po of the oil phase and then computesthe oil saturation So by solving Eq. 22. To obtain thesolution, we use a combination of the implicit-pressure,explicit-saturation (IMPES) and the fully implicitmethods. The limits m = n and m = n + 1 (where n isthe current time step number) of Eqs. 26 or 29 cor-respond, respectively, to the IMPES and fully implicitprocedures. Expressions similar to Eqs. 26 or 29 arealso written down for the eastern, northern, and south-ern neighbors of the base block b , and the resultingterms are all added up to represent the correspondingintegral.

The IMPES method is conditionally stable and con-verges to the correct solution if the time step �t isselected carefully. If �t is too large, the numericalsimulator may generate saturations that are larger thanone. This possibility was eliminated by imposing thecondition that the saturation and pressure changes inany grid block between two consecutive time steps mustremain tightly bounded, which was achieved by adjust-ing �t such that the saturation changes between t andt + �t in any grid block were no more than 0.05, and thepressure changes no more than 100 psi. In addition, toguarantee the stability of the solution, an adaptive time-step method was used which automatically increased �twhenever the variables changed slowly and vice versa.

The advantage of the IMPES method is that, whenused appropriately, it yields accurate results at a min-imum computational cost. Its limitations are that, (a)because a large portion of the reservoir far from thewells may experience very slow changes in Po andSo, it is not efficient to use very small �t, and (b) itis not accurate if large variations in the dependentvariables occur rapidly. The time-truncation errors aregenerally larger in implicit simulators [85, 93]. A fullyimplicit method provides the required stability but at aconsiderably higher computational cost.

The largest variations in So and Po occur in the near-well regions, which also control the maximum allowed

Comput Geosci (2009) 13:187–214 197

time step �t when the flood front is close to them.Therefore, at the beginning of the injection, �t mustbe small, but when the front is no longer near theinjection well, it can be much larger. When the frontreaches the vicinity of the production well(s), �t mustagain be small. Therefore, we consider a small zone ofgrid blocks (of size 7 × 7) around each well in whichwe use the fully implicit procedure for discretizing thegoverning equations, while the IMPES method is usedin the remaining part of the grid.

For the fully implicit part, we guess the So distri-bution at time t and solve the discretized equationsfor Po. The So distribution is then computed based onthe newly calculated Po distribution. The procedureis iterated several times (typically about eight) untilconverged solutions are obtained. For the IMPES part,after computing the Po distribution after each timestep, we solve Eq. 22 in which the time-derivative termis discretized explicitly. In both cases, the discretizedequations are solved by a combination of the Newton–Raphson and biconjugate-gradient methods.

To obtain accurate solutions for Po and So, onemust improve the approximations that are made forestimating the upstream flow functions, which requiresaccurate evaluations of the phase mobilities. Hartenand coworkers [35, 36, 103] introduced a method forevaluating the phase mobilities, usually referred to asthe total variation diminishing (TVD) method, which isfree of the instabilities.

It is increasingly being used in reservoir simulation[76, 98, 102], especially for simulating miscible displace-ments where numerical dispersion may completelydominate the physical dispersion [58, 68, 84, 86, 92].The method requires an appropriate front detectiontechnique which can be based on the calculation ofsuccessive mobility ratios in the neighboring blocks.Higher-order TVD methods are constructed by intro-ducing a variable mobility at each block face given by

λo,face = λo,up + 1

2η1+1/2

(λo,down − λo,up

), (30)

where ηi+1/2 is called the mobility limiter. Here,λo,face = λi+1/2, λo,up = λi, and λo,down = λi+1. Moreover,

ri+1/2 = λo,up − λo,up−1

λo,down − λo,up. (31)

Here, λo,up−1 denotes the oil mobility of up-side block,attached to the upstream block of the base cell. ri+1/2

varies in such a way that the scheme is, at least spatially,second order. The results obtained by using Eq. 30 areusually free of spurious oscillations. Several expressions

for ηi+1/2 have been developed [75, 91, 94]. One of themost accurate limiters [94] is given by,

ηi+1/2 = max

{min

[2ri+1/2,

1

2

(ri+1/2 + 1

), 2

], 0

}. (32)

The above TVD method is applicable to FD-basedsimulations that use uniform grids. According to Eq. 32,in addition to the upstream and downstream mobilities,second-neighbor mobilities λo,up−1 enter the formula-tion as well. Since in a nonuniform grid, the base blockmight have several second-neighbor upstream blocks,we use the maximum mobility among such blocks as therepresentative of the upside mobility; that is, λo,up−1 =λi−1,max. Such a choice guarantees that the approachingfront will be detected by at least one of the neighboringblocks of the base block.

Let us point out that due to its multiscale natureof the wavelets, no assumption needs to be made re-garding the significance, or lack thereof, of the capillarypressure. Clearly, since one obtains a grid with variousblock sizes, the significance of Pc is automatically di-minishes in the large ones.

6 Results and discussion

We have carried out extensive simulations of waterflooding in systems with various geometries and per-meability fields. All the computations were carried outusing a single 3.2 GHz Pentium IV processor. Whatfollows are the description of the model reservoirs andthe discussions of the results.

Case 1 We consider a 2D waterflood in a quarter ofa five-spot system in which the wells are lo-cated diagonally at the opposite corners of thegrid. The geological model is represented by a512 × 512 square grid. The permeability fieldis isotropic and is generated by a FBM withH = 0.2, which corresponds to a heteroge-neous field with negative correlations (largepermeabilities are neighbors to small ones,and vice versa). The semivariogram of a FBMis given by

γ (r) = γ1r2H, (33)

where r is the lag, and γ 1 = γ (r = 1). Equation 33indicates that the semivariogram of a FBM shouldcontinue to increase as the lag r increases (i.e., γ (r)does not exhibit any sill), since the correlation lengthis as large as the system’s size, and no cutoff scalefor the correlations was introduced in the FBM. Theminimum and maximum block permeabilities, (Kmin,

198 Comput Geosci (2009) 13:187–214

Kmax), are (1 mD, 1,620 mD) with a mean of about735 mD (the standard deviation of logK is about 1.0).The porosity was assumed to be constant everywhere.Figure 3 presents the upscaled permeability map, to-gether with the corresponding grid where the small-est blocks represent the geological model’s resolution.The upscaled grid was generated with the thresholdsεs = εd = 0.8. The number of grid blocks was re-duced from the initial 262,144 to only 2,464, with lowcoarsening levels in the high-permeability zones, high

coarsening levels in the low-permeability regions, andno upscaling at all around the wells (see above).

Water is injected at one corner of the grid at aconstant pressure of 6,000 psi, and the fluid is producedat constant total flow rate of 50 bbl/day at the oppositecorner. The relative permeability model used is a con-ventional Corey-type function,

krj(S j

) = k0, j

(S j − Srj

S j,max − Srj

)n j

, (34)

Fig. 3 Case 1 upscaledpermeability map and thecorresponding computationalgrid. The smallest blocksrepresent the resolution ofthe original geological model

Comput Geosci (2009) 13:187–214 199

with K0,o = K0,w = 1, So,max = Sw,max = 0.8, Sro = Srw =0.2, nw = 1.5, and no = 2. The oil and water viscositiesare taken, respectively, to be 5 and 1 cP, and gravityand capillary pressure effects are ignored. Moreover,we took, Cw = 3 × 10−6 and Co = 4 × 10−6, both inpsi−1. We emphasize that it neither poses any particulardifficulty, nor does it increase the method’s computa-tional cost to use many different expressions for Krj(S j)

for different sectors of the grid. Figure 4 comparesthe Sw distributions 175 days after water injection intothe reservoir, computed using both the geological andupscaled models. All the main features of the floodpattern in the geological model are captured by theupscaled grid. However, due to coarsening, the advanc-ing finger becomes smoother in the upscaled model but

causing only a very small delay in the breakthroughtime in the production well. The delay effect can beseen in Fig. 5 where we compare the water flow rates atboth wells in both the geological and upscaled models.These results indicate that, in the upscaled grid, thetime at which the rate of water production has risenfrom zero has a small delay compared to that of thegeological model. Despite this, the trends in the waterflow rate curves of the two models are in very goodagreement.

Since the regions around the wells retain their re-solved structures in the upscaled model, and a fullyimplicit model is used in such regions to solve thegoverning equations (the IMPES formulation is uti-lized in the rest of the reservoir), we may expect close

Fig. 4 Distribution of watersaturation after 175 days ina the geological model andb in the upscaled model

200 Comput Geosci (2009) 13:187–214

0

10

20

30

40

50

60

70

80

0 50 100 150 200 250 300 350

Geo-IWGeo-PWUS-IWUS-PW

Wat

er F

low

Rat

e (b

bl/d

)

Time (days)

Fig. 5 Comparison of the water flow rates at the injection (I) andproduction (P) wells, using the geological (Geo) and the upscaled(US) models

agreement between the two sets of results in terms ofthe bottom-hole pressures. Figure 6 confirms this ex-pectation, where we show the computed well pressuresusing both the geological and the upscaled models. Thesmall difference between the two computed pressuresat the production well is related to the correspondingdifference between the breakthrough times of the twomodels; the difference is about 2%.

Case 2 In this case, the permeability field was strati-fied, with the layers being more or less parallelto the macroscopic flow, generated by an FBMwith H = 0.2 (and setting a1 = 1 in Eq. 21).The statistics of the permeability fields aresimilar to the first case (except that, due tostratification, the field is anisotropic), as is thesize of the grid representing the geologicalmodel. Uniform porosity was assumed every-where. In one case, the grid was upscaled using

0

1000

2000

3000

4000

5000

6000

7000

0 50 100 150 200 250 300 350

Geo-IWGeo-PWUS-IWUS-PW

Bott

om

-Hole

Pre

ssu

re (

psi

)

Time (days)

Fig. 6 Comparison of the bottom-hole pressures in the geologicaland upscaled models. The notations are the same as in Fig. 5

εs = εd = 1.0, which corresponds to uniformcoarsening of the grid. For the second case,εs = εd = 0.8 were utilized. Moreover, dif-ferent levels of upscaling (the parameter l inEqs. 17 and 18) were also utilized in order toassess their effect. The upscaled permeabilitymap and the corresponding grid for εs = εd =0.8 and three coarsening levels (l = 3) areshown in Fig. 7. Note, in particular, the finestructure of the computational grid at the in-terfaces between neighboring layers as well asin the sectors with high permeabilities.

We simulated a water-flooding process in which wa-ter was injected into the reservoir on the left side ofthe grid, at all the grid blocks, at a constant pressureof 6,500 psi, and oil and water were produced from theopposite side at a constant pressure of 5,000 psi (equalto the initial pressure). No-flow boundary conditionwas imposed in the lateral direction. To impose severecondition on the simulations, we simulated a viscosityratio, M = 100. Similar to Case 1, we neglected thecapillary pressure. Figure 8 compares the water flowrates at the outlet face, computed both in the geolog-ical and upscaled models. We simulated the processin several different grids in which the blocks near theinlet and outlet of the system were upscaled at severaldifferent levels, ranging from zero (that is, the blockswere not upscaled) to one and two levels of upscaling(that is, one or two sweepings of the grid; see Section 3).As can be seen, very little difference between all thefigures can be discerned. In addition, the values ofthe two thresholds and whether or not one upscales thegrid blocks near the system’s inlet and outlet appear tomake very little difference. The absence of any effectof the resolution of grid blocks near the boundaries isdue to the absence of the capillary pressure, making theend effects unimportant. We come back to this pointshortly.

Figure 9 compares the computed water saturationsSw(x,y) after 400 days in the geological and upscaledmodels. Once again, the agreement between the twofigures is excellent. Figure 10 compares the time-dependence of the average reservoir pressures in thegeological and upscaled models with three differentlevels of coarsening. All the important features of thecurves agree quantitatively. Figure 11 makes the sametype of comparison, but for the injected flow rates ofwater. Once again, all the cases agree completely witheach other. Figure 12 compares the flow rates of theproduced water in the geological model and those in theupscaled grids, one uniformly coarsened with εs = εd =1.0 and one level (l = 1) of upscaling (which reduces

Comput Geosci (2009) 13:187–214 201

Fig. 7 Case 2 upscaledpermeability map and thecorresponding computationalgrid. The smallest blocksrepresent the resolution ofthe original geological model

the number of the grid blocks by more than 98%) anda second one with three levels of upscaling and εs =εd = 0.8. Once again, the agreement is excellent.

Case 3 This case was similar to Case 2, except that thecapillary pressure was not neglected, and wasrepresented by

Pc = σ

√ϕ

K

(Sw − Swi

1 − Swi − Sor

)4

, (35)

with σ = 20 mN/m being the interfacial tension, ϕ = 0.3the porosity (assumed to be constant everywhere,although it can vary from block to block, if need be),and Swi and Sor the irreducible water saturation and oilresidual saturation. Note, however, that any capillarypressure curve, as well as different Pc curves for differ-ent zones of the grid, may be used. Let us also pointout that the shape of the capillary pressure, which de-pends on the permeability K, has a strong effect on thesaturation distribution and the computational stiffness

202 Comput Geosci (2009) 13:187–214

Fig. 8 Comparison of thewater flow rates in theanisotropic model producedat the outlet (right) face ofthe system if Pc is neglected.The results are for a thegeological model and(εd, εsl) = b (1.0,1.0,2);c (1.0,1.0,1); d (1.0,1.0,0);e (0.8,0.8,2), and f (0.8,0.8,1),where l is the level ofupscaling of the boundarynodes. Note that εs = εd = 1.0corresponds to uniformcoarsening of the grid

of the governing equations that we solve in order tocompute the distribution. In particular, a vanishing, ora mildly varying, capillary pressure does not give riseto end effects that are usually seen in the simulationsof two-phase flow with a strong capillary pressure (seebelow). Moreover, due to the capillary discontinuity,capillary end-effects appear at the system’s inlet andoutlet boundaries, giving rise to larger Sw in theseregions than in the rest of the grid. To capture this effectaccurately, the size of the blocks adjacent to the grid’sleft- and right-most boundaries can be held at the levelof the geological model.

Figure 13 compares the local water flow rates at theoutlet in the geological model with those computed in

the upscaled models for two values of the thresholds εd

and εs and one or two coarsening levels l.To understand the results, we must consider the

variations of Pc with the permeability K. In the geo-logical model the permeability varies from about 1 to1,340 mD. Thus, according to Eq. 35, Pc ≤ 1,200 psi.In general, large capillary pressures tend to imbibewater into the low-permeability regions and smoothenthe front. However, upscaling the blocks reduces thecapillary forces in two important ways:

(1) The higher the blocks’ permeabilities and thesaturation contrasts between adjacent blocks,the higher is the capillary force between them.

Comput Geosci (2009) 13:187–214 203

Fig. 9 Waterssaturation distribution after 400 days, if Pc is ne-glected, in the a anisotropic geological model and b upscaledmodel

5820

5840

5860

5880

5900

5920

5940

5960

5980

0 50 100 150 200 250 300 350 400

Mea

n R

eser

vo

ir P

ress

ure

(p

si)

Geo

l=1,

l=2,

l=3,

l=3,

l=2,

.1

ds

.1

dsε

ε

ε

ε

ε

ε

ε

ε

ε

ε

.1

ds

8.0

ds

8.0

ds

Time (days)

= =

= =

= =

= =

= =

Fig. 10 Time-dependence of the mean reservoir pressure inthe anisotropic geological (Geo) and several upscaled models.Capillary pressure was neglected

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 50 100 150 200 250 300 350 400

Geo

Inje

cted

Wate

r F

low

Rate

(b

bl/

d)

l=1,

l=2,

l=3,

l=3,

l=2,

.1

ds

.1

dsε ε

ε ε

ε ε

ε ε

ε ε

.1

ds

0.8

ds

0.8

ds

Time (days)

= =

= =

= =

= =

= =

Fig. 11 Time-dependence of the injected water flow rates inthe anisotropic geological (Geo) and upscaled models. Capillarypressure was neglected

Upscaling, however, smoothens both the perme-ability and saturation contrasts between the ad-jacent blocks and, hence, reduces the capillaryforces between them.

(2) The capillary forces are inversely proportionalto the blocks’ sizes, implying that one, two, andthree levels of upscaling reduce, respectively, thecapillary forces by factors of 1/2, 1/4, and 1/8. Thenet effect is the weakening of the role of capillaryforces in smoothening the saturation profile. Con-sequently, one has a far more heterogeneous sat-uration distribution and water production history,as the grid is increasingly upscaled (see Fig. 8).

Such expectations are confirmed by the resultsshown in Fig. 13 and their comparison with those shownin Fig. 8. Figure 13d presents the case in which theblocks in a thin region near the left- and right-most

0

10

20

30

40

50

60

0 50 100 150 200 250 300 350 400

Geo

Prod

uced

Wat

er F

low

Rat

e (b

bl/d

)

Time (days)

l=1,.01

dsε ε= =

l=3,.80

dsε ε= =

Fig. 12 Water flow rates in the anisotropic geological (Geo)model and in the upscaled grids, one uniformly coarsened (εs =εd = 1.0) with one level of upscaling (l = 1) and a second one withthree levels of upscaling. Capillary pressure was neglected

204 Comput Geosci (2009) 13:187–214

Fig. 13 a–f Comparison ofwater flow rates for Case 3model, with the capillarypressure, given by Eq. 35,included in the simulations.The notations are the same asthose in Fig. 8

boundaries were not upscaled. The agreement betweenthis case and the geological model, shown in Fig. 13a, isexcellent. However, in contrast with Fig. 8 (for which Pc

was neglected), all the other cases in which the blocksnear the boundaries were upscaled at various levelscould not reproduce the results obtained with the geo-logical model, hence, demonstrating (a) the well-knownend effect and (b) the accuracy of the upscaling method.These results also point out the significance of nonuni-form upscaling. Uniform upscaling (εs = εd = 1.0)of the blocks (except perhaps those on the inlet and out-let faces) results in excessive averaging and smoothen-ing of the heterogeneities (the permeabilities), hence,resulting in delayed breakthrough times. Therefore, the

wavelet thresholds must be smaller than one. One theother hand, upscaling of the blocks at the system’s inletand outlet distorts the true relative significance of thecapillary and viscous forces (recall that the capillaryforces in the larger blocks are smaller). Thus, the op-timal upscaling scenario, for the case in which the cap-illary pressure is significant, is one in which the blocksare coarsened nonuniformly (the wavelet thresholds εs

and εd are smaller than unity), but those at the system’sinlet and outlet are not upscaled. The same is truewhen there are injection and production wells in thesystem.

Figure 14 compares the distributions of Sw in thegeological model and the optimal upscaled model (cor-

Comput Geosci (2009) 13:187–214 205

Fig. 14 a, b Same as in Fig. 9 but with the capillary pressure,given by Eq. 35, included. The results correspond to panel (d) ofFig. 13

responding to Fig. 13d) after 400 days. The capillaryend-effect should give rise to large values of Sw atboth ends, which has been captured accurately by theupscaled model. To make a more quantitative compar-ison between the geological and upscaled models, wepresent in Fig. 15 the computed time-dependence ofthe mean reservoir pressure, indicating very little dif-ference between all the cases (the difference is at most0.2%). At the same time, we should keep in mind thatthree levels of upscaling (l = 3 in Eqs. 17 and 18) greatlyreduces the computation time (see Table 1). Figure 16makes a comparison between the produced water flowrates in the geological and upscaled models, both withuniform (εs = εd = 1.0) and nonuniform upscalingand various levels l of coarsening. As discussed above,

5520

5540

5560

5580

5600

5620

5640

5660

5680

0 50 100 150 200 250 300 350 400

Mea

n R

eser

vo

ir P

ress

ure

(p

si)

Geo

l=1,

l=2,

l=3,

l=3,

l=2,

.1

ds

.1

dsε ε

.1

ds

8.0

ds

8.0

ds

Time (days)

= =

ε ε= =

ε ε=

ε ε= =

ε ε= =

=

Fig. 15 Same as in Fig. 10 but with the capillary pressure, givenby Eq. (3), included

nonuniform upscaling results in excellent agreementbetween the upscaled and geological models. Note thateven one upscaling level reduces the total number ofgrid blocks from 262,144 in the geological model toabout 104 in the nonuniformly upscaled model, result-ing in a speed-up factor of about 60.

Case 4 To test the upscaling scheme under more strin-gent conditions than those considered above,we carried out water-flooding simulations in aquarter of a nine-spot system. The permeabil-ity distribution was generated by FBM with aHurst exponent, H = 0.2, with (Kmin, Kmax) =(2 mD, 1,400 mD), with a stratified structure,as shown in Fig. 17. The relative permeabili-ties were given by Eq. 34, while the capillarypressure was of the Corey type, given by

Pc = 10

[(Sw − Swc

1 − Swc − Sor

)−1/5− 1

]

, (36)

where Swc is the connate water saturation. The poros-ity was assumed to be 0.3. The initial pressure was6,000 psi, and the initial water saturation was assumedto be 0.2. Water was injected at one corner at a con-stant rate of 150 bbl/day, the bottom-hole pressure atthe producing wells was assumed to be 4,500 psi, and300 days of the operations were simulated. The rest ofthe parameters were the same as those used in Case 1.After four and five levels of upscaling, the number ofblocks decreased from the initial 262,144 to 5,632 and3,073, respectively. The resulting speed-up factors inthe computations (see also below) were 958 and 6,203,respectively. This case is of particular interest, as thereare three producing wells in the system (see Fig. 17),

206 Comput Geosci (2009) 13:187–214

Table 1 Comparison of the computation times between those of the geological model (top row) and the upscaled grids and the resultingspeed-up factors (SF)

ε l Grid size t(Pc= 0) SF (Pc= 0) t(Pc = 0) SF (Pc = 0)

262,144 48.74 h 49.05 h1.0 3 1,600 98 s 1,790 81 s 2,1800.8 3 3,499 440 s 398 334 s 5281.0 2 2,368 270 s 652 122 s 1,4470.8 2 4,267 758 s 231 408 s 4321.0 1 3,904 867 s 202 251 s 703

ε = εs = εd refers to the wavelet thresholds, l indicates the level of upscaling, while the grid size refers to the number of blocks inthe grid

such that the flow of the fluids toward them can beparallel or perpendicular to the layers, or be at 45◦ withrespect to them.

Figure 18 compares the saturation distributions after150 days, in both the geological and upscaled models,with the latter one obtained after four levels of up-scaling. The agreement between the two is excellent.Figure 19 compares the water production rates at thethree producing wells, computed with the geological

0

10

20

30

40

50

60

70

80

0 50 100 150 200 250 300 350 400

l=1,

l=2,

l=3,

Pro

du

ced

Wate

r F

low

Rate

(b

bl/

d)

Geo

.01

dsε ε

ε ε

ε ε

.01

ds

1.0

ds

Time (days)

0

10

20

30

40

50

60

70

80

0 50 100 150 200 250 300 350 400

Pro

du

ced

Wate

r F

low

Rate

(b

bl/

d)

Time (days)

= =

= =

= =

l=1,

l=2,

l=3,

Geo

.80

.80

.80

dsε ε

ε ε

ε εds

ds

= =

= =

= =

Fig. 16 Same as in Fig. 12 but with capillary pressure, given byEq. (3), included

model, and two upscaled grids obtained after four andfive levels of upscaling. Once again, the agreement in allthe cases is excellent. In particular, the breakthroughtimes are correctly predicted by the upscaled model.Figure 20 makes the same type of comparison thatFig. 19 does, but for the rates of oil production at theproducing wells.

Case 5 As the final test of the upscaling method, wesimulated water flooding in a 2D reservoirin which three intersecting fractures were in-serted, with one of them being connected tothe injection well. The three fractures wereassumed to have a permeability of 20,000 mDand to occupy one line of the grid. This isshown in Fig. 21. Note, in particular, the finestructure of the grid around the interface be-tween the fractures and the matrix. The capil-lary pressure was ignored. Water was injectedat one corner at a rate of 50 bbl/day, and thefluids were extracted at the opposite corner.The rest of the parameters were the same asthose of Case 4. Three levels of upscaling withεs = εd = 0.7, 0.8, and 0.9 were utilized whichreduced the total number of grid blocks to1,415, 1,066, and 775, respectively.

The resulting speed-up factors in the computations(see also below) were, respectively, 269, 455, and 963.Figure 22 compares the saturation distributions after10 days in the geological and the upscaled model af-ter three upscaling levels. The agreement between thetwo is good. To make a more quantitative comparisonbetween the two models, we compare in Fig. 23 therate of oil production at the producing well in both thegeological upscaled models. All the important featuresof the results in the geological model are producedcorrectly by the upscaled models. Moreover, as thelevel of upscaling increases, the agreement between

Comput Geosci (2009) 13:187–214 207

Fig. 17 a Case 4 permeabilitydistribution in the ageological model and b theupscaled grid after four levelsof coarsening. One quarter ofa nine-spot system isconsidered

the computed values in the two models also improves.Figure 24 compares the rates of water production at theproducing well in the geological and upscaled modelswith the same level of agreement as that of Fig. 23.

7 The speed-up of the wavelet upscaling

In addition to its accuracy, any reasonable upscalingmethod must provide significant speed-up in the com-

208 Comput Geosci (2009) 13:187–214

Fig. 18 Water saturation distribution in Case 4 reservoir (seeFig. 17) after 150 days in the a geological model and b upscaledmodel with four levels of upscaling

putations so that one can upscale a large geologicalmodel and obtain accurate results in a reasonable com-putation time. The upscaling method that we developedin this paper is highly efficient for the water-floodingproblem in the quarter of a five-spot system and resultsin huge savings in the computation times. While thesimulations with the geological model took about 475 h(19.8 days), those with the upscaled model took only0.5 h (using the same computer), resulting in a speed-up factor of 950. Using larger wavelet thresholds (thanεs = εd = 0.8 that we used) would result in even largersavings in the computation time.

0

10

20

30

40

50

60

0 50 100 200 250 300150

Wa

ter

Pro

du

ctio

n R

ate

(b

bl/

da

y)

Time (days)

Geo-P1Geo-P2Geo-P3US-L4-P1US-L4-P2US-L4-P3US-L5-P1UP-L5-P2US-L5-P3

Fig. 19 Comparison of the water productions rates in Case 4geological model (Geo) and the upscaled models with four andfive levels of coarsening

The acceptable difference between the results ob-tained with the geological and upscaled models sets thedesired values of the thresholds. In Table 1, we com-pare the CPU times for the first three cases (those forCases 4 and 5 were already mentioned above) for waterflooding in the systems in which water was injectedat one face and oil and water were produced at theopposite face. The minimum speed-up factor that thewavelet-based yields is 202 but is as large as 6,203 (forCase 4, see above)—more than three orders of magni-tude increase in the speed of the computations. We arenot aware of any method that comes close to such greatincreases in the efficiency of the computations while,as the results presented above indicated, preserving theaccuracy of the solutions.

0

5

10

15

20

25

30

35

40

45

0 50 100 150 200 250 300

Geo-P1

Geo-P2

Geo-P3

US-L4--P1

US-L4-P2

US-L4-P3

US-L5-P1

US-L5-P2

US-L5-P3

Oil

Pro

du

ctio

n R

ate

(b

bl/

da

y)

Time (days)

Fig. 20 Same as in Fig. 19 but comparing the oil production rates

Comput Geosci (2009) 13:187–214 209

Fig. 21 a Case 5 permeabilitydistribution with threefractures and b the resultingupscaled grid withεs = εd = 0.8

We should point out an important point regardingthe efficiency of the wavelet method: the broader thepermeability distribution, the more efficient is the waveletmethod. The reason is clear: if the heterogeneities are

distributed only relatively mildly, then significant fluidflow takes place through many sectors of the system. Onthe other hand, if the medium is highly heterogeneous,then only a small fraction of it contributes significantly

210 Comput Geosci (2009) 13:187–214

Fig. 22 Water saturationsafter 10 days in a Case 5geological model and b theupscaled model of Fig. 21

to fluid flow. This important concept, known in physicsof heterogeneous media as the critical-path concept,has proven to be very powerful for a wide varietyof highly disordered media [78, 80]. As a result, thenumber of grid blocks in the upscaled model, when the

heterogeneities are mildly distributed, is larger thanthose in a highly heterogeneous medium.

This concept was recently demonstrated [67] for sim-ulation of frequency-dependent conduction in a 3D het-erogeneous material, in which the local conductivities

Comput Geosci (2009) 13:187–214 211

0

2

4

6

8

10

12

14

16

18

20

0 50 100 150 200

Geo

US-1

US-2

US-3

Oil

Pro

du

ctio

n R

ate

(b

bl/

da

y)

Time (days)

Fig. 23 Same as in Fig. 19 but for Case 5

were distributed over 12 orders of magnitude. Thewavelet method reduced the number of grid blocksfrom 2,097,152 in the initial model to only 630. Thedifference between the effective conductivities of thetwo models, when the frequencies were varied over tenorders of magnitude, was at most 5%. Elsewhere [73],we have demonstrated the same levels of efficiency andaccuracy by simulating water flooding in the SPE-10model [16], in which the permeabilities vary over eightorders of magnitude in a three-dimensional reservoir,exhibiting severe channeling, while the porosities varyover three orders of magnitude, with the fine-scalegrid containing over 1.1 million grid blocks. The highaccuracy of the results presented in the present paperand those for the SPE-10 model testify to the power ofthe method described in this paper. Further tests of themethod, under a variety of conditions, are underway.The results will be reported elsewhere.

0

5

10

15

20

25

30

35

40

45

0 50 100 150 200

Wa

ter

Pro

du

ctio

n R

ate

(b

bl/

da

y)

Time (days)

Geo

US-1

US-2

US-3

Fig. 24 Same as in Fig. 20 but for Case 5

8 Summary

We have developed an accurate and efficient methodfor upscaling of geological models of heterogeneousreservoirs for use in the simulation of two-phase fluidflow. The method is based solely on the wavelettransformation of the single-phase permeabilities; notwo-phase flow property is upscaled. The technique,together with the improvements in the numericalmethods that we suggested for solving the governingequations for two-phase flows, yields very significantspeed-ups in the computations. In particular, a speed-up factor of over 6,000 that we obtain for the Case 4study in the presence of the capillary pressure is ademonstration of the method’s high efficiency. Lohneet al. [54] proposed a two-stage upscaling method fortwo-phase flows from core to simulation scale, arguingthat heterogeneities in the sub-geological scales (cen-timeter to meter scale) may have a significant effect onflow in a reservoir which cannot be captured if the up-scaling starts at the geostatistical scale. They computedthe effective properties for the geological units at thegeostatistical scale by starting the upscaling at a smallscale.

The effective properties at the reservoir simulationscale were obtained from a second upscaling using asthe input the effective properties from the first step.Similar approaches were used earlier by [18, 88] inwhich the effect of very small-scale heterogeneities wasincluded in the generation of the pseudo-functions forvarious geological facies. The wavelet-based methodproposed here greatly reduces the computational costof the two stages. This will be demonstrated in a futurepaper.

Finally, as Case 5 studied above hinted, the proposedmethod may be suited for development of upscaledmodels of fractured reservoirs, where one needs to useresolved grid blocks near the fractures in addition to thehigher-permeability regions of the matrix and the near-well zones. The results with this type of reservoirs willbe reported in the near future.

Acknowledgements The work of M.R.R. was supported by theNIOC. We thank Drs. Fatemeh Ebrahimi, Amir Heidarinasab,and Manouchehr Haghighi for useful discussions.

Open Access This article is distributed under the terms of theCreative Commons Attribution Noncommercial License whichpermits any noncommercial use, distribution, and reproductionin any medium, provided the original author(s) and source arecredited.

212 Comput Geosci (2009) 13:187–214

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