Comput Geosci (2008) 12:257–272DOI 10.1007/s10596-007-9073-7

ORIGINAL PAPER

Multiscale simulations of porous media flowsin flow-based coordinate system

Y. Efendiev · T. Hou · T. Strinopoulos

Received: 2 April 2006 / Accepted: 10 September 2007 / Published online: 6 March 2008© Springer Science + Business Media B.V. 2007

Abstract In this paper, we propose a multiscale tech-nique for the simulation of porous media flows in aflow-based coordinate system. A flow-based coordinatesystem allows us to simplify the scale interaction andderive the upscaled equations for purely hyperbolictransport equations. We discuss the applications of themethod to two-phase flows in heterogeneous porousmedia. For two-phase flow simulations, the use of aflow-based coordinate system requires limited globalinformation, such as the solution of single-phase flow.Numerical results show that one can achieve accurateupscaling results using a flow-based coordinate system.

Keywords Porous media flow ·Flow-based coordinate system · Two-phase flow ·Multiscale · Upscaling

1 Introduction

The modeling of two-phase flow in porous forma-tions is important for both environmental remediationand the management of petroleum reservoirs. Practical

Y. Efendiev (B)Department of Mathematics, Texas A&M University,College Station, TX 77843-3368, USAe-mail: efendiev@math.tamu.edu

T. Hou · T. StrinopoulosApplied Mathematics, Caltech, Pasadena, CA 91125, USA

T. Houe-mail: hou@acm.caltech.edu

T. Strinopoulose-mail: theofilos@acm.caltech.edu

situations involving two-phase flow include the disper-sal of a nonaqueous phase liquid in an aquifer or thedisplacement of a nonaqueous phase liquid by water.In the subsurface, these processes are complicated bythe effects of permeability heterogeneity on the flowand transport. Simulation models, if they are to providerealistic predictions, must accurately account for theseeffects. However, because permeability heterogeneityoccurs at many different length scales, numerical flowmodels cannot in general resolve all of the scales ofvariation. Therefore, approaches are needed for repre-senting the effects of subgrid scale variations on larger-scale flow results. Typically, upscaled or multiscalemodels are employed for such systems. The main ideaof upscaling techniques is to form coarse-scale equa-tions with a prescribed analytical form that may differfrom the underlying fine-scale equations. In multiscalemethods, the fine-scale information is carried through-out the simulation and the coarse-scale equations aregenerally not expressed analytically but, rather, formedand solved numerically.

On the fine (fully resolved) scale, the subsurfaceflow and transport of N components can be describedin terms of an elliptic (for incompressible systems)pressure equation coupled to a sequence of N ! 1hyperbolic (in the absence of dispersive and capillarypressure effects) conservation laws. Our purpose in thispaper is to perform upscaling of two-phase immisci-ble flow in a flow-based coordinate system. The flow-based coordinate system provides us with a solutionthat is better suited for upscaling because the solutionis smoother and the scale interaction can be simpler.We would like to note that single-phase flow upscalingmethods have been employed in the Cartesian frame-work using flow-based grids [25]. In flow-based grid

258 Comput Geosci (2008) 12:257–272

upscaling, the original equations are solved in a flow-based grid generated using the streamlines of theflow. Because of the similarities with the upscaling inflow-based grid, we use the terminology “flow-basedcoordinate system.” The use of flow-based coordinatesystem allows us to perform accurate upscaling of thetransport equation, which is purely hyperbolic. Theupscaling of the transport equation along the stream-lines in the current pressure-streamline coordinatesystem can be obtained because the equation is one-dimensional. We discuss the upscaling of the saturationequation across the streamlines. This type of upscalingintroduces nonlocal macrodispersion terms, where themacrodispersion term involves two-point correlation ofthe velocity field along the streamlines. The advan-tage of using the flow-based coordinate system is thatthe computation of macrodispersion can be performedsemianalytically. This allows us to avoid the typicaldifficulties [13] associated with the computation of themacrodiffusion term, including cross diffusion terms.

For upscaling of the pressure equation, we employmultiscale finite element type methods (MsFEM). Ms-FEM was first introduced in Hou and Wu [16]. Its mainidea is to incorporate the small-scale information intofinite element basis functions and capture their effecton the large scale via finite element computations. Thisapproach shares common features with a number ofother multiscale numerical methods, such as residualfree bubbles [6, 22], variational multiscale method [18],MsFEM [16], two-scale finite element methods [21],and two-scale conservative subgrid approaches [2]. Weremark that special basis functions in finite elementmethods have been used earlier in Babuska and Osborn[3] (cf. [4]). Multiscale finite element methodology hasbeen modified and successfully applied to two-phaseflow simulations in Jenny et al. [19, 20] and Aarnes[1] and later in Chen and Hou [8]. Arbogast [2] usedvariational multiscale strategy and constructed a mul-tiscale method for two-phase flow simulations. Whenconsidering two-phase flow upscaling, we use multi-scale basis functions to compute the macrodispersion.Recently, a limited global information has been used[1, 11] in constructing multiscale basis functions. It isinteresting to note that, in the flow-based coordinatesystem, these multiscale methods reduce to a standardmultiscale finite element method [16].

The paper is organized in the following way: Inthe next section, we present the governing equations.In Section 3, we briefly present a motivation for ourapproach. Section 4 is devoted to the upscaling oftransport equation (hyperbolic equation). In Section 5,we briefly mention multiscale finite element methods.The numerical results are presented in Section 6.

2 Fine-scale equations

We consider two-phase flow in a reservoir ! underthe assumption that the displacement is dominated byviscous effects; i.e., we neglect the effects of gravity,compressibility, and capillary pressure. Porosity willbe considered to be constant. The two phases will bereferred to as water and oil, designated by subscriptsw and o, respectively. We write Darcy’s law, with allquantities dimensionless, for each phase as follows:

v j = !krj(S )

µjk · " P, (1)

where v j is the phase velocity, k is the permeability ten-sor, krj is the relative permeability to phase j ( j = o, w),S is the water saturation (volume fraction), P is pres-sure, and µj is the viscosity of phase j ( j = 0, w). In thiswork, a single set of relative permeability curves is usedand k is assumed to be a diagonal tensor. CombiningDarcy’s law with a statement of conservation of massallows us to express the governing equations in termsof the so-called pressure and saturation equations:

" · ("(S )k · " P) = q, (2)

#S#t

+ v · " f (S ) = 0, (3)

where " is the total mobility, f is the fractional flowof water, q is a source term, and v is the total velocity,which are respectively given by:

"(S ) = krw(S )

µw

+ kro(S )

µo,

f (S ) = krw(S )/µw

krw(S )/µw + kro(S )/µo, (4)

v = vw + vo = !"(S )k · " P. (5)

The above descriptions are referred to as the finemodel of the two-phase flow problem. For simplicity,in further analyses, we will assume q = 0 and imposenonhomogeneous boundary conditions.

3 Motivation

The upscaling and multiscale methods for two-phaseflow systems have been discussed by many authors.In most upscaling procedures, the upscaled quantitiesare computed on a coarse grid obtained from the un-derlying fine-grid. Because of strong scale interactionsassociated with complex spatial correlations, most up-scaling and multiscale methods may require some type

Comput Geosci (2008) 12:257–272 259

of global information or border regions to take intoaccount nonlocal neighboring information. In the pro-posed work, multiscale, and upscaling techniques areused in a flow-based coordinate system. This coordinatesystem, which is based on limited global information,can provide additional smoothness for the solution, andit simplifies the scale interaction.

Next, we present the flow-based coordinate systemand the modified equation and discuss some asymp-totic properties of the modified equations in the flow-based coordinate system. We will restrict our analysisto the two-dimensional case and assume that the het-erogeneous porous medium is isotropic, k(x) = k(x)I.In general, the stream function is defined as " # $ =v = (v1, v2). In two dimensions, the stream function $

reduces to a scalar field defined by

#$/#x1 = !v2, #$/#x2 = v1. (6)

It can be easily shown that "$ · " p = 0, where p isinitial pressure. Denote the initial stream function andpressure by $ = $(x, t = 0) and p = P(x, t = 0). If weassume S = 0 at time zero, then ($, p) can be ob-tained from the pressure equation with "(S ) = 1. Then,the equation for pressure and stream function can bewritten down in this curvilinear orthogonal coordinatesystem using standard coordinate transformation tech-niques (see [11, 23]). As a result, we obtain

#

#$

!k2"(S )

# P#$

"+ #

#p

!"(S )

# P#p

"= 0. (7)

Applying the same change of variables to saturationequation, we get

#S#t

+ (v · "$)# f (S )

#$+ (v · " p)

# f (S )

#p= 0. (8)

Our objective is to present an upscaling methodin pressure-streamline framework for two-phase flowequations. The cornerstone of our upscaling methodis the fact that the pressure at later times is a smoothfunction of initial pressure and, thus, the upscaling ofinitial pressure-streamline framework is more robustand accurate. In Efendiev et al. [11], the authors con-sider a special case when the permeability has strongnonlocal effects with a single high-permeability chan-nel. The authors show that, in this case, the pressureevolution can be written as

P($, p, t) = P0(p, t) + h.o.t, (9)

where P0 is the solution of

#

#p

!"0(p, t)

# P0

#p

"= 0, (10)

where "0 depends S0. If " is a smooth function, then P0

is a smooth function with respect to p. Here, h.o.t arerelated to the contrast in the permeability field, which isvery high inside the channel. This expansion shows that,in porous media with strong channelized nonlocal ef-fects, the initial pressure-streamline coordinate systemcan provide a better coordinate system for performingupscaling. First, this coordinate system can simplify thescale interaction and nonlocal effects can be modeledmore accurately.

4 Upscaling of saturation equation in flow-basedcoordinate system

4.1 Homogenization of saturation equation

In this section, we would like to derive an upscaledmodel for the transport equation. We will assume thatthe velocity is independent of time, "(S ) = 1, andrestrict ourselves to the two-dimensional case. Then,using the pressure-streamline framework, one obtains

S %t + v%

0 f (S %)p = 0

S(p, $, t = 0) = S0, (11)

where % denotes the small scale and v%0 denotes the

Jacobian of the transformation and is positive. Forsimplicity, we assume k(x) = k(x)I and we have "$ ·" p = 0. For deriving upscaled equations, we will firsthomogenize Eq. 11 along the streamlines, and then ho-mogenize across the streamlines. The homogenizationalong the streamlines can be done following Bourgeatand Mikelic [5] or following Hou and Xin [17] andWeinan [10]. The latter uses two-scale convergencetheory, and we refer to Strinopoulos [23] for the re-sults on homogenization of Eq. 11 using two-scaleconvergence theory. We note that the homogenizationresults of Bourgeat and Mikelic are for general het-erogeneities without an assumption on periodicity, andthus, they are more appropriate for problems consid-ered in the paper. Following Bourgeat and Mikelic [5],the homogenization of Eq. 11 can be easily derived(see Proposition 3.4 in [5]).

For ease of notations, we ignore the $ dependenceof v%

0 and S% and treat $ as a parameter. We consider

v%0(p) = v0

#p,

p%

$,

where the velocity field has both large-scale and small-scale variations. Moreover, we assume that the domainis a unit interval. Then, for each $ , it can be shown that

260 Comput Geosci (2008) 12:257–272

S%(p, $, t) $ S(p, $, t) in L1((0, 1) # (0, T)), where Ssatisfies

St + v0 f (S )p = 0, (12)

and where v0 is harmonic average of v%0 , i.e.,

1v%

0$ 1

v0weak % in L&(0, 1),

as % $ 0. The proof of this fact follows from Proposi-tion 3.4. of Bourgeat and Mikelic [5]. Here, we brieflysketch the proof.

Following Bourgeat and Mikelic [5] and assuming forsimplicity

% 10

d&v%

0(&)=

% 10

d&v0(&)

= 1, we introduce

dX%(p)

dp= v%

0(X%(p)),dX0(p)

dp= v0(X0(p)).

Then (Lemma 3.1 of Bourgeat and Mikelic [5]):

X% $ X0 in C[0, 1] as % $ 0. (13)

Consequently,

& T

0

& 1

0|S%(p, ' ) ! S(p, ' )|dpd'

=& T

0

& 1

0|S%(X%(p), ' )! S(X%(p), ' )|v%

0(X%(p))dpd'

'& T

0

& 1

0|S%(X%(p), ' )! S(X0(p), ' )|v%

0(X%(p))dpd'

+& T

0

& 1

0|S(X%(p), ' )! S(X0(p), ' )|v%

0(X%(p))dpd'

'& T

0

& 1

0|S%(X%(p), ' ) ! S(X0(p), ' )|dpd'

+& T

0

& 1

0|S(X%(p), ' ) ! S(X0(p), ' )|dpd' (14)

The first term on the right-hand side of Eq. 14 con-verges to zero because S%(X%(p), ' ) and S(X0(p), ' )

satisfy the same equation ut + f (u)p = 0, however, withthe following initial conditions S%(X%(p), t = 0) = S0 (X% S(X0(p), ' ) = S0 ( X0. Because of Eq. 13 and com-parison principle& T

0

& 1

0|S%(X%(p), ' ) ! S(X0(p), ' )|dpd'

' C& 1

0|S0 ( X% ! S0 ( X0|dp,

the first term converges to zero. The convergence of thesecond term for each $ follows from the argument in

Bourgeat and Mikelic ([5], page 368) using Lebesgue’sdominated convergence theorem.

Next, we provide a convergence rate (see also [23])of the fine saturation S % to the homogenized limit S as% $ 0.

Theorem 1 Assume that v%0(p) is bounded uniformly

C!1 ' v%0

#p,

p%

$' D.

Denote by F(t, T) the solution to St + f (S )T = 0. Thesolution S of Eq. 12 converges to S % (assuming initialconditions that don’t depend on the fast scale) at a rategiven by

)S % ! S)& ' G%,

when F remains Lipschitz for all time, and

)S % ! S)n ' G%1/n,

when F develops at most a finite number of dis-continuities.

The proof of this theorem is provided in theAppendix (see also [23]). We remark that T, whichis time-of-flight, is used as an independent variable.We briefly show the results of a numerical exper-iment to demonstrate the estimate of Theorem 1for a discontinuous solution. We consider a non-linear flux f (S ) with µo = µw = 1, T final = 0.1, v =1 ! 20p sin

#10( 1

p+0.1

$+ 20sin

'5(

2

(. Note that we only

solve the hyperbolic equation, and Riemann initial con-dition is considered. To find the rate of convergenceof S to S % , we have to use a grid that resolves thevelocity and the shock so that numerical error, es-pecially the numerical diffusion near the shock, doesnot mask the upscaling error. We use a grid of 4,096cells for both upscaled and fine computation to avoidthe discretization errors as much as possible. At thesame time, the velocity must vary enough in the cellsso that the upscaling error is large. The top row ofTable 1 shows the number of coarse grid blocks. Toavoid numerical diffusion, we use a small final time.The upscaling errors are shown in Table 1. The L&norm shows that, in all experiments with less than 64coarse cells, the numerical diffusion was not significant.The convergence rate seems to be slightly larger than 1,which is consistent with Theorem 1.

The homogenized operator given by Eq. 12 still con-tains variation of order % through the fast variable $

%;

however, there, it does not contain any derivatives inthat variable. Its dependence on $

%is only parametric.

We can homogenize the dependence of the partially ho-mogenized operator on $

%and arrive at a homogenized

Comput Geosci (2008) 12:257–272 261

Table 1 Numericaldemonstration of Theorem 1

2 4 8 16 32 64 128

L1 0.0432 0.0124 0.0063 0.0049 0.0022 6.62 # 10!4 1.41 # 10!4

L& 0.670 0.664 0.663 0.665 0.653 0.61 0.096

operator that is independent of the small scale. In thelatter case, we will only obtain weak convergence ofthe partially homogenized solution. When we homog-enized along the streamlines, the resulting equationwas of hyperbolic type, like the original equation. In aseminal and celebrated paper, Tartar [24] showed thathomogenization across streamlines leads to transportwith the average velocity plus a time-dependent diffu-sion term, referred to as macrodispersion, a physicalphenomenon that was not present in the original fineequation. In particular, if the velocity field does notdepend on p inside the cells, that is, v($, $

%), then the

homogenized solution, S (weak% limit of S, which willbe denoted by S ), satisfies

St + v0Sp =& t

0

&Spp(p ! "(t ! ' ), $, ' )dµ $

%(")d'.

(15)

Here, d)$%

the Young measure associated with thesequence v0($, ·) and dµ $

%is a Young measure that

satisfies)& d)$

%(")

s2( iq + "

*!1

= s2( iq

+ v0 !& dµ $

%(")

s2( iq + "

.

We have denoted by v0 the weak limit of the velocity.This equation has no dependence on the small scale,and we consider it to be the full homogenization of thefine saturation equation. Efendiev and Popov [14] haveextended this method for the Riemann problem in thecase of nonlinear flux. Note that the homogenizationacross streamlines provides a weak limit of partiallyhomogenized solution. Because the original solution S%

strongly converges to partially homogenized solutionfor each $ , it can be easily shown that S% $ S weakly.We omit this proof here.

In numerical simulations, it is difficult to use Eq. 15as a homogenized operator, and often, a second-orderapproximation of this equation is used. These approxi-mate equations can also be derived using perturbationanalysis. In particular, using the higher moments ofthe saturation and the velocity, one can model themacrodispersion. In the context of two-phase flow, thisidea was introduced by Efendiev et al. [12, 13]. Inour case, the computation of the macrodispersion ismuch simpler because the transport equations have

been already averaged along the streamlines, and thus,we will be applying the perturbation technique to one-dimensional problem.

We expand S, v0 (following [13]) as an averageover the cells in the pressure-streamline frame and thecorresponding fluctuations

S = S(p, $, t) + S*(p, $, t)

v0 = v0(p, $, t) + v*0(p, $, t). (16)

We will derive the homogenized equation for f (S ) =S. Averaging Eq. 12 with respect to $ , we find anequation for the mean of the saturation

St + v0Sp + v*0S*

p = 0.

An equation for the fluctuations is obtained by sub-tracting the above equation from Eq. 12

S*t + (v0 ! v0) Sp + v0S*

p ! v*0S*

p = 0.

Together, the equations for the saturation are

St + v0Sp + v*0S*

p = 0

S*t + v*

0Sp + v0S*p ! v*

0S*p = 0. (17)

We can consider the second equation to be the aux-iliary (cell) problem and the first equation to be theupscaled equation. We remind that the cell problem fora hyperbolic equation is O(1), whereas for an elliptic,it is O(%). We can obtain an approximate numericalmethod by solving the cell problem only near the shockregion in space time, where the macrodispersion termis largest. In that case, it is best to diagonalize theseequations by adding the first to the second one

St + v0 Sp = !v*0(Sp ! Sp)

St + v0 Sp = 0.

Compared to Eq. 17, it has fewer forcing terms and nocross fluxes, which leads to a numerical method withless numerical diffusion that is easier to implement. Inmost cases, we can make a better approximation, whichis described in the following sections.

4.2 Comparison with the cell problemin the cartesian frame

Before we derive the numerical approximation, we willbriefly compare the homogenized equations in flow-based frame with the homogenized equations in a

262 Comput Geosci (2008) 12:257–272

Cartesian frame obtained recently in Hou et al. [15]and Westhead [26]. The homogenized equations in theCartesian variables as derived in Hou et al. [15] andWesthead [26] are defined in terms of the average satu-ration over the coarse blocks S and the fluctuations S**.Note that, whereas the fluctuations S* in the pressure-streamline frame depend only on one fast variable,the fluctuations S** in the Cartesian frame depend ontwo fast variables. P is a projection operator onto theaverage along the streamlines within the cell, which cor-responds to the fast variable along the streamlines, andQ is a projection onto the orthogonal complement, sothat any function u can be written as u = P(u) + Q(u).With this notation, the homogenized equations are

St + v · "S + " · v**S** = 0

S**t + (v + P(v**)) · "S** + P(v**) · "S

! " · v**S** = G#

x,x%, t

$, (18)

where

G#

x,x%, t

$= (v + P(v**)) · Q("S**)

!P(Q(v**) · Q("S**)) + Q(S**).

The Cartesian cell problem, which is the equation forthe fluctuations in Eq. 18, is a two-dimensional equationalong two fast variables. Prior to solving it, one mustcompute the projections P and Q, which adds to thecomplexity of the method and its computational cost.In contrast, the pressure-streamline cell problem inEq. 17 contains only one fast variable and no projectionoperator. In some sense, in the pressure-streamlineframe, the projection operation, which was carried outby restricting the oscillatory test functions, removes afast variable and reduces one fast dimension to arriveat the cell problem of Eq. 17. In the Cartesian frame,the projection operation remains in the equations in theform of P and Q and the fast variation along the flow isnot cleanly removed. This is a strong indication that thepressure-streamline frame reveals the structure of theflow correctly.

4.3 Numerical averaging across streamlines

The derivation in the previous sections contained noapproximation. In this section, we follow the sameidea as in the derivation to solve the equation forthe fluctuations along the characteristics, but with thepurpose of deriving an equation on the coarse grid. Toachieve this, we will not perform analytical upscalingin the sense of deriving a continuous upscaled equationas in the previous section. We will first discretize the

equation with a finite volume method in space and thenupscale the resulting equation. Our upscaled equationwill therefore be dependent on the numerical scheme.

We use the same definition for the average satura-tion and the fluctuations as in Eq. 16 and follow thesame steps until Eq. 17. We discretize the macrodisper-sion term in the equation for the average saturation

v*0S*

p = v*0S*i+1 ! v*

0S*i

*p+ O(*p).

A superscript ·i refers to a discrete quantity defined atthe center of the conservation cell. Instead of solvingthe equation for the fluctuations on the fine characteris-tics as before, which would lead to a fine grid algorithm,we solve it on the coarse characteristics defined by

dPdt

= v0, with P(p, 0) = p.

Compared to the equation that we obtained in theprevious section for S*, this equation for S* has an extraterm, which appears second

S* = !& t

0

#v*

0(P(p, ' ), $)Sp(P(p, ' ), $, ' )

+ v*0(P(p, ' ), $)S*

p(P(p, ' ), $, ' ) + v*0S*

p)$

d'.

The second term is second-order in fluctuating quanti-ties, and we expect it to be smaller than the first term sowe neglect it. As before, we multiply by v*

0 and averageover $ to find

v*0S* = !

& t

0v*

0v0(P(p, ' ), $)Sp(P(p, ' ), $, ' )d'.

In this form at time t, it is necessary to know infor-mation about the past saturation in (0, t) to computethe future saturation. Following [13], it can be easilyshown that Sp(P(p, ' ), $, ' ) depends weakly on time,in the sense that the difference between Sp and Sp is ofthird-order in fluctuating quantities. Therefore, we cantake Sp out of the time integral to find

v*0S* = !

& t

0v*

0v*0(P(p, ' ), $)d' Sp.

The term inside the time integral is the covariance ofthe velocity field along each streamline. The macrodis-persion in this form can be computed independent ofthe past saturation.

Comput Geosci (2008) 12:257–272 263

The nonlinearity of the flux function introduces anextra source of error in the approximation. We expandf (S ) near S (cf. [12]) and keep only the first term

S = S(p, $, t) + S*(p, $, t)

v0 = v0(p, $, t) + v*0(p, $, t)

f (S ) = f (S ) + fS(S )S* + O(S*2)

f (S )p = fS(S )Sp + f (S )S* + . . . (19)

This approximation is not accurate near the shockbecause S* is not small near sharp fronts. The regionnear the shock is important because the macrodisper-sion is large. Due to the dependence of the jump in thesaturation on the mobility, we expect this approxima-tion to be better for lower oil mobilities. Nevertheless,this approximation works well in practice. For moreaccuracy, it is also possible to retain more terms inthe Taylor expansion. We will show that, in realistic

examples, these higher-order terms are not importantin our setting.

Using these definitions, we derive the followingequations for the average saturation and the fluctua-tions (see [23] for more details)

St + v0 f (S )p + v*0( fS(S )S*)p = 0 (20)

S*t + v*

0 fS(S )Sp + v0 fS(S )S*p ! v*

0S*p = 0. (21)

The macrodispersion is discretized as

v*0( fS(S )S*)p = v*

0 fS(S )S*i+1

! v*0 fS(S )S*

i

*p+ O(*p).

We solve the second equation on the coarse character-istics defined by

dPdt

= v0 fS(S ), with P(p, 0) = p

and form the terms that appear in the macrodispersion

v*0 fS(S )S* = !

& t

0v*

0 fS(S )v*0(P(p, ' ), $) fS(S(P(p, ' ), $, ' ))Sp(P(p, ' ), $, ' )d'.

As before, we have dropped terms that are second-order in fluctuating quantities. It can be shown (see[23]) that fS(S(P(p, ' ), $, ' ))Sp(P(p, ' ), $, ' ) doesnot vary significantly along the streamlines, and it canbe taken out of the integration in time:

v*0 fS

'S

(S* = !

& t

0v*

0v*0(P(p, ' ), $)d' fS

'S

(2Sp. (22)

This expression is similar to the one obtained in thelinear case; however, the macrodispersion depends onthe past saturation through the equation for the coarsecharacteristics.

Even though the macrodispersion depends on thepast saturation, it is possible to compute it incremen-tally, as it is done in Efendiev and Durlofsky [12]. Givenits value D(t) at time t, we compute the values at t + *tusing the macrodispersion at the previous time

D(t + *t) =& t+*t

0. . . d' =

& t

0. . . d' +

& t+*t

t. . . d'.

This is possible because, in the derivation for the ap-proximate expression for the macrodispersion, we tookthe terms that depend on S(' ) outside the time inte-gration. The integrand, the average covariance of thevelocity field along the streamlines, needs to be com-puted only once at the beginning. Then, updating the

macrodispersion takes O(n2) computations, as many asit takes to update S.

5 Multiscale finite element methods for upscalingof pressure equation

The multiscale finite element framework will be usedfor upscaling of the pressure equation. We will choosea finite volume element method as a global solver.The proposed method is similar to an earlier intro-duced multiscale finite volume method in Jenny et al.[19, 20]. Let Kh denote the collection of coarse ele-ments/rectangles K. Consider a coarse element K, andlet ! K be its center. The element K is divided intofour rectangles of equal area by connecting ! K to themidpoints of the element’s edges. We denote thesequadrilaterals by K! , where ! + Zh(K ) are the verticesof K. Also, we denote Zh = +

K Zh(K ) and Z 0h , Zh

the vertices that do not lie on the Dirichlet boundary of!. The control volume V! is defined as the union of thequadrilaterals K! sharing the vertex ! .

The key idea of the method is the construction ofbasis functions on the coarse grids, such that thesebasis functions capture the small-scale information oneach of these coarse grids. The basis functions areconstructed from the solution of the leading order

264 Comput Geosci (2008) 12:257–272

homogeneous elliptic equation on each coarse elementwith some specified boundary conditions. Thus, if weconsider a coarse element K that has d vertices, thelocal basis functions +i, i = 1, · · · , d are set to satisfy thefollowing elliptic problem:

!" · (k · "+i) = 0 in K, +i = gi on #K, (23)

for some function gi defined on the boundary of thecoarse element K. In previous findings, the functiongi for each i is chosen to vary linearly along #K or tobe the solution of the local one-dimensional problems[19] or the solution of the problem in a slightly largerdomain is chosen to define the boundary conditions[16]. We will be using linear boundary conditions andrequire +i(x j) = ,ij. Finally, a nodal basis function asso-ciated with the vertex xi in the domain ! is constructedfrom the combination of the local basis functions thatshare this xi and zero elsewhere.

Next, we denote by Vh the space of our approxi-mate pressure solution, which is spanned by the basisfunctions {+ j}x j+Z 0

h. A statement of mass conserva-

tion on a coarse-control volume Vx is formed from(2), where now the approximate solution is writtenas a linear combination of the basis functions. As-sembly of this conservation statement for all con-trol volumes would give the corresponding linearsystem equations that can be solved accordingly. To bespecific, the problem now is to seek P h + Vh with P h =,

x j+Z 0h

P j+ j such that&

#V!

"(S )k · " P h · n dl = 0, (24)

for every control volume V! , !. Here, n defines thenormal vector on the boundary of the control volume,#V! , and S is the fine-scale saturation field at this point.

In the previous studies [1, 7, 11], it was found that theuse of global information can improve the multiscalefinite element method. In particular, the solution of the

pressure equation at initial time is used to constructthe boundary conditions for the basis functions. It isinteresting to note that the multiscale finite elementmethods that employ a limited global information re-duce to standard multiscale finite element methods in aflow-based coordinate system. This can be verified di-rectly, and the reason behind it is that we have alreadyemployed a limited global information in a flow-basedcoordinate system.

6 Numerical results

In this section, we first show representative simulationresults for "(S ) = 1 for flux functions f (S ) = S andnonlinear f (S ) with viscosity ratio µo/µw = 5. Forsuch a setting, the pressure and saturation equationsare decoupled and we can investigate the accuracy ofsaturation upscaling independently from the pressureupscaling. We note that this decoupling of the pressureand saturation equations is artificial. At the end ofthe section, we will present numerical results for two-phase flow with variable "(S ). We consider two typesof permeability fields. The first type includes a per-meability field generated using two-point geostatisticswith correlation lengths lx = 0.3, lz = 0.03, and - 2 =1.5 (see Fig. 1, left). The second type corresponds toa channelized system, and we consider two examples.The first example (middle figure of Fig. 1) is a syntheticchannelized reservoir generated using both multipointgeostatistics (for the channels) and two-point geostatis-tics (for permeability distribution within each facies).The second channelized system is one of the layersof the benchmark test (representing the North Seareservoir), the Society of Petroleum Engineers compar-ative project [9] (upper Ness layers). These permeabil-ity fields are highly heterogeneous, channelized, anddifficult to upscale. Because the permeability fields are

-1

0

1

2

3

4

5

6

7

8

9

-4

-2

0

2

4

6

8

-3

-2

-1

0

1

2

3

Fig. 1 Permeability fields used in the simulations. Left, permeability field with exponential variogram; middle, synthetic channelizedpermeability field; right, layer 36 of the Society of Petroleum Engineers comparative project [9]

Comput Geosci (2008) 12:257–272 265

highly heterogeneous, they are refined to 400 # 400 toobtain accurate comparisons.

Simulation results will be presented for saturationsnapshots and the oil cut as a function of pore volumeinjected (PVI). Note that the oil cut is also referred to asthe fractional flow of oil. The oil cut (or fractional flow)is defined as the fraction of oil in the produced fluid andis given by qo/qt, where qt = qo + qw, with qo and qw

being the flow rates of oil and water at the productionedge of the model. In particular, qw =

%#!out f (S )v · ndl,

qt =%#!out v · ndl, and qo = qt ! qw, where #!out is the

outer flow boundary. We will use the notation Q fortotal flow qt and F for fractional flow qo/qt in numericalresults. PVI, defined as PV I = 1

Vp

% t0 qt(' )d' , with Vp

being the total pore volume of the system, provides adimensionless time for the displacement.

When using multiscale finite element methods fortwo-phase flow, one can update the basis functionsnear the sharp fronts. Indeed, sharp fronts modify thelocal heterogeneities, and this can be taken into ac-

count by resolving the local equations, Eq. 23, for basisfunctions. If the saturation is smooth in the coarseblock, it can be approximated by its average in Eq. 23,and consequently, the basis functions do not need tobe updated. It can be shown that this approximationyields first-order errors (in terms of coarse mesh size).In our simulations, we have found only a slight im-provement when the basis functions are updated; thus,the numerical results for the MsFEM presented in thispaper do not include the basis function update near thesharp fronts. Because a pressure-streamline coordinatesystem is used, the boundary conditions are given byP = 1, S = 1 along the p = 1 edge and P = 0 along thep = 0 edge, and no flow boundary condition on the restof the boundaries.

For the upscaled saturation equation, which is aconvection–diffusion equation, we need to observe anextra Courant–Friedrichs–Lewy (CFL)-like conditionto obtain a stable numerical scheme *t ' *p2

2), where

) is the diffusivity. In our case, the diffusivity is

0.1

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0.5

0.6

0.7

0.8

0.9

0.1

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0.3

0.4

0.5

0.6

0.7

0.8

0.9

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0.7

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0.8

0.9

Fig. 2 Saturation snapshots for variogram based permeability field (top) and synthetic channelized permeability field (bottom). Linearflux is used. The left figures represent the upscaled saturation plots, and the right figures represent the fine-scale saturation plots

266 Comput Geosci (2008) 12:257–272

0.1

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0.7

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0.9

Fig. 3 Saturation snapshots for variogram based permeability field (top) and synthetic channelized permeability field (bottom). Non-linear flux is used. The left figures represent the upscaled saturation plots, and the right figures represent the fine-scale saturation plots

%cell

% t0 v*

0(p(' ), $)v*0(p, $)d'd$ . If the macrodispersion

is large, this can be a very restrictive condition. Toremedy this, we used an implicit discretization for themacrodispersion. This is straightforward because theproblem is one-dimensional. The resulting system wassolved by a tridiagonal solver very fast. Because theorder of the highest derivative in the equation hasincreased, we require extra boundary conditions. Forthe computation of the macrodispersion term, we im-pose no flux on both boundaries of the domain.

In the upscaled algorithm, a moving mesh is usedto concentrate the points of computation near thesharp front. Because the saturation equation is one-dimensional in the pressure-streamline coordinates, theimplementation of the moving mesh is straightforwardand efficient. For the details, we refer to Strinopoulos[23]. We compare the saturation right before the break-through time so that the shock front is largest. For thiscomparison, we also average the fine saturation overthe coarse blocks because the upscaled model is defined

Table 2 Upscaling error forpermeability generated usingtwo-point geostatistics

25 # 25 50 # 50 100 # 100 200 # 200

Linear fluxL1 error of S 0.0021 6.57 # 10!4 2.15 # 10!4 8.75 # 10!5

L1 error of S with macrodispersion 0.115 0.0696 0.0364 0.0135L1 error of S fine without macrodispersion 0.1843 0.0997 0.0505 0.0191

Nonlinear fluxL1 error of S 0.0023 8.05 # 10!4 2.89 # 10!4 1.29 # 10!4

L1 error of S with macrodispersion 0.116 0.0665 0.0433 0.0177L1 error of S fine without macrodispersion 0.151 0.0805 0.0432 0.0186

Comput Geosci (2008) 12:257–272 267

Table 3 Upscaling errorfor synthetic channelizedpermeability field

25 # 25 50 # 50 100 # 100 200 # 200

Linear fluxL1 error of S 0.0222 0.0171 0.0122 0.0053L1 error of S with macrodispersion 0.0819 0.0534 0.0333 0.0178L1 error of S fine without macrodispersion 0.123 0.0834 0.0486 0.0209

Nonlinear fluxL1 error of S 0.0147 0.0105 0.0075 0.0040L1 error of S with macrodispersion 0.0842 0.0658 0.0371 0.0207L1 error of S fine without macrodispersion 0.119 0.0744 0.0424 0.0214

Table 4 Upscaling errorfor SPE10, layer 36 25 # 25 50 # 50 100 # 100 200 # 200

Linear fluxL1 error of S 0.0128 0.0093 0.0072 0.0042L1 error of S with macrodispersion 0.0554 0.0435 0.0307 0.0176L1 error of S fine without macrodispersion 0.123 0.0798 0.0484 0.0258

Nonlinear fluxL1 error of S 0.0089 0.0064 0.0054 0.0033L1 error of S with macrodispersion 0.0743 0.0538 0.0348 0.0189L1 error of S fine without macrodispersion 0.0924 0.0602 0.0395 0.0202

Table 5 Total error forpermeability field generatedusing two-point geostatistics

25 # 25 50 # 50 100 # 100 200 # 200

Linear fluxL1 upscaling error of S 0.0021 6.57 # 10!4 2.15 # 10!4 8.75 # 10!5

L1 error of S computed on coarse grid 0.0185 0.0062 0.0019 0.0015L1 upscaling error of S 0.115 0.0696 0.0364 0.0135L1 error of S computed on coarse grid 0.139 0.0779 0.0390 0.0144

Nonlinear fluxL1 upscaling error of S 0.0023 8.05 # 10!4 2.89 # 10!4 1.29 # 10!4

L1 error of S computed on coarse grid 0.0268 0.0099 0.0027 9.38 # 10!4

L1 upscaling error of S 0.116 0.0665 0.0433 0.0177L1 error of S computed on coarse grid 0.146 0.0797 0.0461 0.0184

Table 6 Total error forsynthetic channelizedpermeability field

25 # 25 50 # 50 100 # 100 200 # 200

Linear fluxL1 upscaling error of S 0.0222 0.0171 0.0122 0.0053L1 error of S computed on coarse grid 0.0326 0.0161 0.0107 0.0113L1 upscaling error of S 0.0819 0.0534 0.0333 0.0178L1 error of S computed on coarse grid 0.135 0.0849 0.0477 0.0274

Nonlinear fluxL1 upscaling error of S 0.0147 0.0105 0.0075 0.0040L1 error of S computed on coarse grid 0.0494 0.0295 0.0150 0.0130L1 upscaling error of S 0.0842 0.0658 0.0371 0.0207L1 error of S computed on coarse grid 0.17 0.11 0.0541 0.0303

268 Comput Geosci (2008) 12:257–272

Table 7 Total errorfor SPE10 layer 36 25 # 25 50 # 50 100 # 100 200 # 200

Linear fluxL1 upscaling error of S 0.0128 0.0093 0.0072 0.0042L1 error of S computed on coarse grid 0.023 0.0095 0.0069 0.0052L1 upscaling error of S 0.0554 0.0435 0.0307 0.0176L1 error of S computed on coarse grid 0.0683 0.052 0.0361 0.0205

Nonlinear fluxL1 upscaling error of S 0.0089 0.0064 0.0054 0.0033L1 error of S computed on coarse grid 0.0338 0.0148 0.0074 0.0037L1 upscaling error of S 0.0743 0.0538 0.0348 0.0189L1 error of S computed on coarse grid 0.115 0.0720 0.0406 0.0204

on a coarser grid. In Figs. 2 and 3, we plot the saturationfor linear and nonlinear (with µo/µw = 5) f (S ). Weremind that "(S ) = 1 in these simulations. As we seein both cases, we have very accurate representation ofthe saturation profile.

We proceed with a quantitative description of the er-ror. We will distinguish between two sources of errors.We will refer to the difference between the upscaledand the fine-scale solution as the upscaling or modelingerror. In this case, the upscaled equations are solved onthe fine grid to avoid the discretization errors on thecoarse grid. We note that the discretization errors, suchas the ones associated with a numerical diffusion, canbe very large on relatively coarse grids. We will refer tothe difference between the fine-scale solution and thesolution of the upscaled equation on the coarse grid asthe total error. In this case, the upscaled equations aresolved on the coarse grid and the total error includesboth the upscaling error and the discretization error onthe coarse grid. To compute the upscaling error, wecompare the upscaled solution computed on a 400 #400 grid with the fine-scale saturation computed onthe 400 # 400 grid and averaged over the coarse grid.The errors are computed in the p, $ frame and arerelative errors. We display the upscaling error againstthe number of coarse cells in Tables 2, 3, and 4. Linearflux and nonlinear (µo/µw = 5) flux cases are consid-ered with "(S ) = 1. In these tables, S refers to the

solution upscaled along streamlines (see Eq. 12) and Srefers to the solution upscaled both along and acrossthe streamlines (see Eq. 21). We see from this table thatupscaling using macrodispersion reduces the upscalingerrors. Note that the effects of macrodispersion aremore significant in the case of linear flux when thejump discontinuity in the saturation profile is larger. Weremark that the macrodispersion errors become lesssignificant as the coarse mesh size decreases. Becausethe sharp fronts are resolved in our numerical simula-tions, the macrodispersion represents the fluctuationsof the solution away from these fronts. The effects ofthese fluctuations decrease as the coarse mesh becomessmaller.

In Tables 5, 6, and 7, we show the total error, thatis, the modeling and discretization error, for the casesconsidered in Tables 2, 3, and 4. We again remind that Sis the solution upscaled along streamlines (see Eq. 12),S is the solution upscaled both along and across thestreamlines (see Eq. 21). The first and third rows inthese tables are the errors computed on the fine grid,while the second and fourth rows represent the errorscomputed on the coarse grid. First, we note that thetotal errors are much smaller when the upscaling is onlyperformed along the streamlines. It is interesting thatthe convergence of S to S is observed even though theupscaling error is larger than the numerical error ofthe fine solution. The reason is that the location of the

Table 8 Computational costFine x.y Fine p, $ S S

Layered, linear flux 5648 257 9 1Layered, nonlinear flux 14543 945 28 4Percolation, linear flux 8812 552 12 1Percolation, nonlinear flux 23466 579 12 1SPE10 36, linear flux 40586 1835 34 2SPE10 36, nonlinear flux 118364 7644 25 2

Comput Geosci (2008) 12:257–272 269

Fig. 4 Left: pressure and streamline function at time t = 0.4 in Cartesian frame. Right: pressure and streamline function at time t = 0.4in initial pressure-streamline frame

moving mesh points was selected so that the points areas dense near the shock as the fine solution using theparameter hmin. This was done to observe the upscalingerror clearly and also to have similar CFL constraintson the time step, which allows a clean comparison ofcomputational times. We compare the require CPUtimes in Table 8. We note that it took 26 units oftime to interpolate one quantity from the Cartesian tothe pressure-streamline frame. The upscaled solutionswere computed on a 25 # 25 grid, and the fine solutionwas computed on a 400 # 400 grid, so we expect theS computations to be 256 times faster or more. Theextra gain comes from a less restrictive CFL conditionbecause we use an averaged velocity. The computationsin the Cartesian frame are much slower.

The application of the proposed method to two-phase immiscible flow can be performed using theimplicit pressure and explicit saturation (IMPES)framework. This procedure consists of computing thevelocity and then using the velocity field in updatingthe saturation field. When updating the saturation field,we consider the velocity field to be time-independent,and we can use our upscaling procedure at each IMPEStime step. First, we note that, in the proposed method,the mapping is done between the current pressure-streamline and initial pressure-streamline. This map-ping is nearly the identity for lower oil mobilities. InFig. 4, we plot the level sets of the pressure and streamfunction at time t = 0.4 in a Cartesian coordinatesystem (left plot) and in the coordinate system of the

0.1

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0.8

0.9

Fig. 5 Left: Saturation plot obtained using coarse-scale model. Right: The fine-scale saturation plot. Both plots are on coarse grid.Variogram based permeability field is used. µo/µw = 5

270 Comput Geosci (2008) 12:257–272

Fig. 6 Comparison offractional flow for coarse-and fine-scale models.Variogram-basedpermeability field is used.µo/µw = 5

0 0.5 1 1.5 2 2.5 3 3.5 4pvi

0 0.5 1 1.5 2 2.5 30

0.1

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0.5

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0.7

0.8

0.9

1

time

fract

iona

l flo

w ra

te

fineupscaled

initial pressure and streamline (right plot). Clearly,the level sets are much smoother in initial pressure-streamline frame compared to Cartesian frame. Thisalso explains the observed convergence of upscalingmethods as we refined the coarse grid. In Fig. 5, we plotthe saturation snapshots right before the breakthrough.In Fig. 6, the fractional flow is plotted. Again, the mov-ing mesh algorithm is used to track the front separately.The convergence table is presented in Table 9. We seefrom this table that the errors decreases as first order,which indicates that the pressure and saturation aresmooth functions of initial pressure and streamline.

Table 9 Convergence of the upscaling method for two-phaseflow for variogram based permeability field

50 # 50 100 # 100 200 # 200

With SL2 pressure error at t = 3T final

4 0.0014 0.0007 0.0004

L2 velocity error at t = 3T final4 0.0235 0.0137 0.0072

L1 saturation error t = T final 0.0105 0.0052 0.0027With S

L2 pressure error at t = 3T final4 0.0046 0.0021 0.0008

L2 velocity error at t = 3T final4 0.0530 0.0335 0.0246

L1 saturation error t = T final 0.0546 0.0294 0.0134

7 Conclusions

In this paper, multiscale methods for two-phase im-miscible flow using flow-based coordinate system areconsidered. In particular, the upscaling of a convection-dominated transport equation is discussed. The flow-based coordinate system allows us to simplify the scaleinteraction and obtain an upscaled model for transport.Furthermore, this upscaled model is used to designa coarse-scale algorithm for two-phase flow. In ournumerical methods, the shock front of the upscaledequation was resolved using a moving mesh. Numericalresults show that one can achieve high accuracy usingthe proposed algorithms. The proposed methods areefficient when the solution is smooth in a new coordi-nate system (perhaps with the exception of some sharpmoving fronts). However, if, under changing sourceterms or boundary conditions, the solution dramaticallychanges and no longer remains smooth, then one mayneed to introduce another coordinate system for accu-rate coarse-scale simulations.

Though the results presented in the paper are en-couraging, there are possible extensions that are cur-rently under investigation. The extension to threedimensions does not seem to be difficult. However,it fails when the coordinate transformation becomesdegenerate. We conjecture (see [23]) that, for mostpermeability fields, the regions of the flow where this

Comput Geosci (2008) 12:257–272 271

occurs do not carry much fluid. It should then be pos-sible to regularize the transformation and apply thismethod with only a small numerical error. Another di-rection of future research is the development of fast andaccurate algorithms that perform interpolation fromthe Cartesian to pressure-streamline grid. In particular,our interest is in the development of such algorithms us-ing coarse-scale information similar to multiscale finiteelement methods. The latter will speed-up the interpo-lation computations and make the method more desir-able for multiphase flow and transport computations.

Acknowledgements We would like to thank Dr. Victor Gintingfor providing us with the multiscale finite volume code. We wouldlike also to thank Dr. Yuguang Chen for providing us with thesynthetic channelized permeability fields and the reviewers fortheir valuable comments. This research is supported by DOEgrant DE-FG02-06ER25727. T.Y.H. would like also to acknowl-edge partial support from the NSF Information TechnologyResearch grant ACI-0204932 and the NSF Focused ResearchGroups grant DMS-0353838.

Appendix

Proof of Theorem 1

First, we note that the velocity bound implies thatC!1 ' v0(p) ' D, uniformly. We transform the equa-tions for S % Eq. 11 and S Eq. 12 to the time-of-flightvariable defined by

dT%

dp= 1

v%0(p, $)

T%(0)= 0for S % and

dTdp

= 1

v#

p, $, $%

$

T(0)= 0

for S.

Both equations reduce to

St + f (S )T = 0.

The solution to this equation is F(t, T). Because theinitial condition does not depend on %, neither doesF. Then, S = F(t, T%(P, .)), S = F(t, T(P, .)). Usingthese expressions for the saturation, we can obtainthe desired estimates by following the same steps asin the linear case. When F remains Lipschitz for alltimes we can easily obtain a pointwise estimate in termsof the Lipschitz constant M )S % ! S)& = )F(t, T%) !F(t, T))& ' M)T% ! T)& ' G%. Otherwise, we willneed the time-of-flight bound that we derived for thelinear flux that reduces here to

|T%(P) ! T(P)| ' 2C%. (25)

We will divide the domain in regions where F isLipschitz with constant M in the second variable, de-noted by A2, and shock regions, denoted by A1, and

estimate the difference of S % and S in each region sepa-rately. To fix the notation, let there be n discontinuitiesin F(t, ·) of magnitude less than *F, which does nothave to be small, at {T = Ti}i=1,...,n. We will denotethe thin strips of width 2C% around the discontinuitieswith A1

A1 ={T such that |T!Ti| ' 2C%, for some i=1,. . ., n}

and with A2 its complement. We selected the width ofthe strip based on Eq. 25, so that for any point P, ifT%(P) /+ A1, then T%(P) and T(P) are on the same sideof any jump Ti. When T%(P) + A2, F is Lipschitz in theregion between T% and T, and we can show%

A2(S % ! S)2dpd$ =

%A2

(F(t, T%) ! F(t, T))2dpd$

' M2)T% ! T)2&|T%(A2)

!1|' N2%2|T%(A2)

!1|,

where we used the time-of-flight bound Eq. 25. By|T%(A2)

!1|, we denoted the image of A2 under the in-verse of T%(P). Inside the strip A1, even though S % andS differ by an O(1) quantity, we can use the smallnessof the area of the strip to make the L2 norm of theirdifference small%

A1(S % ! S )2dpd$ =

%A2

(F(t, T%) ! F(t, T))2dpd$

' (*S + N%)2|T%(A1)!1|

' (*S + N%)24CDn%.

We estimated the area |T%(A1)!1| by using the def-

inition of A1 and the fact that the Jacobian of thetransformation T%(P)!1 is v%

0 and is bounded uniformlyin p, $ . Putting together the two estimates for regionsA1 and A2, we obtain )S % ! S)2 ' G%1/2. Estimates interms of the other Lp norms follow similarly.

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