A discontinuous Galerkin method for two-phase ﬂow i

8/13/2019 A discontinuous Galerkin method for two-phase ow i

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A discontinuous Galerkin method for two-phase

flow in a porous medium enforcing H(div) velocity

and continuous capillary pressure

Todd Arbogast, Mika Juntunen, Jamie Pool, and Mary F. Wheeler

April 16, 2012

Abstract

We solve the slightly compressible two-phase flow problem in aporous medium with capillary pressure. The problem is solved usingthe implicit pressure, explicit saturation method (IMPES), and theconvergence is accelerated with iterative coupling of the equations.We use discontinuous Galerkin to discretize both the pressure and thesaturation equation. We propose two improvements, namely projectingthe flux to the mass conservativeH(div)-space and penalizing the jumpin capillary pressure in the saturation equation. We also discuss theneed and use of slope limiters, and the choice of primary variables in

discretization. The methods are verified with numerical examples. Theresults show that the proposed modifications stabilize the method andimprove the solution considerably.

1 Introduction

We consider the problem of slightly compressible two-phase flow in a porousmedium, which is the simplest equation for subsurface flow in carbonatereservoir modeling. Solving this problem effectively and accurately is req-uisite for solving the more elaborate models of, e.g., CO2 sequestration orenhanced oil recovery [5]. We assume that the each phase has its own pres-

sure, so there is capillary pressure between the phases. Taking this effect intoaccount is essential for the validity of the model in subsurface applications[9]. We describe our problem in more detail in section 2.1.

In this article we solve the system using the implicit pressure, explicitsaturation method (IMPES) [4]. In this approach the problem is decoupledto pressure and saturation equations, and then solved sequentially. To re-duce the error of decoupling, and thereby allow larger time steps, we useiterative coupling to solve the pair of decoupled equations. The effectivnessof this iterative coupling was demonstrated in, e.g., [6, 12, 11, 10]. We recallthe IMPES algorithm in section 2.2. We also discuss the delicate issue of

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choosing the reference pressure for the problem, a choice which turns out to

have a profound effect on the solution when capillary pressure is present.We choose to discretize the equations using discontinuous Galerkin (DG)

for both of the decoupled equations. This DG-DG-method is briefly de-scribed in section 3, see, e.g., [11, 7] for more detailed derivation. Anotherpopular method is to solve the pressure equation using mixed finite elements(MFE), and use DG only for the saturation equation. For more details onthe MFE-DG-method, see, e.g., [13, 2]. In section 3.2 we show how theflux coming from the DG-DG approach can be post processed into a locallymass conservativeH(div)-space flux, imitating the MFE-DG approach. Thispost processing stabilizes the saturation part of the problem in our DG-DGapproach.

In section 3.3, we propose a new technique for maintaining the continuityof capillary pressure. We introduce a penalty for the jump in capillary pres-sure in the saturation equation. In the case of only one capillary pressurecurve throughout the domain, this forces the saturation to be continuous,which is physically correct. In the case of multiple capillary pressure curves,this improvement allows the saturation to jump at the interface betweendistinct capillary pressure curves, and instead enforces the physically rele-vant continuity of capillary pressure. This method is able to capture thediscontinuity of saturation even when the absolute difference in capillarypressures is small. Our approach has similarities with [8].

Finally, in section 4 we demonstrate the numerical performance of our

proposed method using several examples. The examples are computed witha research code developed at the Center for Subsurface Modeling [1] of theUniversity of Texas at Austin.

2 Problem formulation

In this section we recall the two-phase Darcy problem modeling subsurfaceflow and the IMPES algorithm used for solving the non-linear equations.The presentation is brief; for more details see, e.g., [4].

2.1 The continuous problemOur model problem is for slightly compressible two-phase flow. The massconservation equation states that

iSit

+ ui = fi, i= w, n, (1)

in which is the porosity, i is the density, Si is the saturation, ui is theflux,fi is the source/sink, and subscriptsw and n stand for the wetting and

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non-wetting phases respectively. Darcys law gives

ui =kiKi

i(pi igD), (2)

in whichki is the relative permeability, K is the permeability, i is the vis-cosity, pi is the pressure, g is the gravity, and D defines the direction ofgravity (being the gradient of depth D). The equations are coupled nonlin-early through relative permeability, capillary pressure, and density. Relativepermeability is a known function of saturation

ki = ki(Sw). (3)

The capillary pressure function models the pressure difference between thewetting and non-wetting phases. It is also a known function of saturation

pc = pnpw = pc(Sw). (4)

We assume slight compressibility, so

i = i(pi) =ref,iecipi, (5)

in whichci is the known compressibility coefficient and ref,i is the referencedensity.

The model assumes that all the pores are filled with fluid,

Sw+Sn = 1, (6)

and in what follows, we assume that porosityand viscosityiare constantsfor simplicity of exposition.

2.2 The IMPES algorithm with iterative coupling

The IMPES algorithm decouples the equations and solves them sequentially.For completeness we present the IMPES equations below. The first step isto sum the mass conservation equations (1) for both phases

(uw+ un) =fw+fn wSwt + nSnt . (7)In practice, the mass conservation equations (1) are scaled with the referencedensityref,i. If we suppose for the moment that the densities were constant,equation (6) shows that the last term above disappears, i.e.,

wSwt

+ nSn

t = 0, (8)

and hence (uw+ un) =fw+fn. (9)

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With the slight compressibility assumption, the last term in (7) is indeed

very small and is often neglected. For simplicity, we will assume in thederivation that this term vanishes. However, we emphasize that in ournumerical examples, this term is present.

Next we modify (9) by using Darcys law (2) and the definition of cap-illary pressure (4) to obtain

(tKpw+nKpc ()tgKD) =ft (10)

in which i = kii/i, t = w +n, ()t = ww +nn, and ft =fw + fn. The subscriptt stands for total. Above we have chosen the wettingphase pressure pw to be the primary variable. We will discuss the choice ofreference pressure in more detail at the end of this section.

Assume we know the solution of the pressure and saturation at time tk,that is, we know pkw andS

kw. To compute the pressure p

k+1w at timet

k+1 wetime lag the saturation to compute ki and p

kc , and time lag pressure to get

densityki . Rearranging the terms gives

(kt Kpk+1w ) =ft+ (

knKp

kc ()

kt gKD), (11)

in which all the unknowns are on the left and the right hand side in known.This is the equation for the implicit pressure step of the IMPES algorithm,although we will modify it to the form (13) later.

Once we have the wetting phase pressure at time tk+1, we solve for

the saturation at time tk+1 using an explicit time stepping in the massconservation equation (1)

k+1w Sk+1w

t =fw u

k+1w +

kwSkw

t , (12)

in which t = tk+1 tk and the velocity uk+1w is computed using Darcyslaw (2). How to compute the velocity properly is discussed in more detailin section 3.2. The previous equation (12) is the explicit saturation step ofthe IMPES algorithm.

2.3 Enhancing the IMPES algorithm

In this section we discuss enhancements to the IMPES algorithm to make itmore effective in practice. The enhancements are iterative coupling, takingshorter saturation steps, and choosing the reference pressure.

Iterative coupling. The IMPES algorithm decouples the non-linear equa-tions and solves them sequentially; hence, the results are sometimes inaccu-rate. There may be a loss of mass balance and a mismatch in the acquiredpressure and saturation. If either of these is detected a simple improvementis to iterate by redoing the time step using the new pressure and saturationin equation (11). The pseudo code for this algorithm is:

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1. Givenpkw andSkw, solve for p

k+1k using (11);

2. Givenpk+1w and Skw, solve for S

k+1w using (12);

3. Check for convergence; and if converged, continue to next time step,and otherwise resetpk+1w p

kw and S

k+1w S

kw, and go to step 1.

In the numerical examples presented in this article, we always use this iter-ative coupling technique.

Shorter saturation steps. In reservoir simulation it is often the case thatthe saturation part of the IMPES algorithm has much more rapid changesthan the pressure part. In fact, the pressure is almost independent of timeif the wells are operated with constant pressure or constant flow. Therefore

it is well-known to be useful to take shorter time steps in the saturationequation. For example, if the time step in the pressure equation is t,one can take N saturation time steps of length t/Nbefore taking a newpressure step. In the intermediate steps, the pressure is interpolated linearly.Taking several saturation steps can speed up the simulation considerablysince the saturation equation is explicit, which is much easier to solve thanthe implicit pressure equation. In the numerical examples presented in thispaper we have used roughly 10 saturation steps for each pressure step. Thishas been a favorable ratio in terms of how long it takes to run the simulation.

Choosing the reference pressure. In the IMPES algorithm describedabove we have used the wetting phase pressure and saturation as the refer-

ence variables. The choice between the reference saturations seems irrelevantby (6). One can simply choose the saturation that is easiest to implement.However the choice of reference pressure is a more delicate issue. If onechooses the non-wetting phase pressure as the reference pressure, equation(11) becomes

(kt Kpk+1n ) =ft+ (

kwKp

kc ()

kt gKD). (13)

The difference between equations (11) and (13) is in the capillary pres-sure term, involving knp

kc for the wetting phase and

kwp

kc for the

non-wetting phase reference pressure. Near the residual saturation for the

wetting phase, that is, at Sw,r , the relative permeability for the wettingphase vanishes, kw(Sw,r) = 0, and the relative permeability for the non-wetting phase assumes its highest value, kn(Sw,r) 1. At the same timethe capillary pressure, however, tends to infinity, pc(Sw,r) . Combin-ing these observations, shows that the term knp

kc , needed when using the

wetting phase pressure as reference, tends to infinity. However, the termkwp

kc , needed when using the non-wetting phase as reference pressure,

stays bounded. The conclusion is that, numerically, it is better to use thenon-wetting phase pressure as the reference pressure. We will study theeffect of reference pressure with numerical examples in section 4.

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3 Discretization

3.1 DG-DG two-phase flow

We present the discrete form of the problem (1)-(6) in this section, in theform of (11) and (12) using the discontinuous Galerkin, (DG), method. Fora more detailed derivation of DG methods for flows in porous media see,e.g., [11, 7] and the citations therein.

Assume we are solving the problem in the domain that has piecewisesmooth boundary . For simplicity, we assume the boundary is dividedinto three non-overlapping pieces: in denotes the inflow boundary whereboth pressure pinw and saturation S

inw are defined, out denotes the outflow

boundary where only pressure poutw needs to be given, and0 denotes the

no-flow boundary where the total normal flow is zero.We assume the domain is divided into a set of non-overlapping ele-

mentsE, and the collection of elements, the mesh, is denoted by Th, whereh denotes the maximum diameter of the elements. We assume the mesh isquasi-uniform, i.e. the mesh diameter is bounded from below. We denote byGh the collection of edges/faces of the mesh Th. The subsetG

inth denotes the

internal edges, and the subsets Ginh , Gouth , and G

0h denote the inflow, outflow

and no-flow boundary edges, respectively.The approximation spaces for the solution are

Wl ={v L2() :v|E Pl(E) , E Th}, (14)

meaning that we use piecewise polynomials of order l and the space is dis-continuous between elements. In the numerical examples, we use space W1

for pressure and either W0 or W1 for the saturation.On an edge e between elements E1 andE2,{{v}}= (v1+ v2)/2denotes

the average,[[v]] =v1 v2 denotes the jump,denotes the outer normal ofelementE1. On a boundary edge,denotes the outer normal of the domain.

Now we are ready to form the weak form of the pressure step (11). Find

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pk+1w Wl, l 1 such that

ETh

kt Kp

k+1w , q

E

+

eGinth

{{kt Kpk+1w }}, [[q]]

e

{{kt Kq }}, [[p

k+1w ]]

e

+

he

[[kt Kp

k+1w ]], [[q]]

e

+

eGin

hGout

h

kt Kpk+1w , q

e

kt Kq , p

k+1w

e

+

he

kt Kp

k+1w , qe

=ETh

ft, q

E

knKpkc ()

kt gKD, q

E

+

eGint

h

{{(knKp

kc ()

kt gKD) }}, [[q]]

e

+

eGinhGout

h

knKp

kc

kt

gKD

, qe

(15)

+eGin

h

kt Kq , p

inw

e+

he

kt Kp

inw , q

e

+ eGout

h

kt Kq , poutw e+

he kt Kpoutw , qe (16)for all q Wl, > 0 is the stability parameter, K =

T K is thepermeability in the normal direction, and parameter = 1, 0, 1defines themethod. The three values of theta correspond to NIPG, IIPG, and SIPG1,respectively.

Formally, the weak form for the saturation step (12) is: FindSk+1w Wm,

m 0, such that

EThk+1w S

k+1w

t , zE

=ETh

kwS

kw

t , z

E

+

fw, zE

uk+1w , zE

(17)

for all z Wm. Since we only solve for the wetting phase pressure, pk+1w ,we do not directly have the wetting phase velocity uk+1w . Instead, we needto construct the velocity using Darcys law (2). To this end, we integrate

1The acronyms stand for non-symmetric (N), incomplete (I), and symmetric (S) interiorpenalty Galerkin (IPG).

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by parts in the last term to obtain

ETh

k+1w S

k+1w

t , z

E

=ETh

kwS

kw

t , z

E

+

fw, zE

+uk+1w , z

E

eGint

h Gin

hGout

h

uk+1w , ze. (18)

Inside the elements we can use Darcys law, but on the edges e we need totake into account the discontinuities, that is, the numerical flux. On internaledges, the flux is computed as

uk+1w = (

kwK)

up{{(pk+1w k+1w gD) }} + (

w

t)up

he[[pk+1w ]], (19)

in which ()up means that the quantities are upwinded. Upwinding meansthat on an edge e, the values are computed using the data of the outflowelement. On in/outflow boundaries the flux is computed in similar manner;the difference is that we do not need to use the mean value, and that the jumpis computed against the given boundary data. Using (19), the saturationstep (17) becomes

ETh

k+1w S

k+1w

t , z

E

(20)

= ETh

kwSkwt

, zE

+ fw, zE+ (kwK)up(pk+1w k+1w gD), zE

eGint

h

(kwK)

up{{(pk+1w k+1w gD) }} +

wt

up he

[[pk+1w ]], ze

eGin

h

(kwK)

up(pk+1w k+1w gD) +

wt

up he

(pk+1w pinw ), z

e

eGouth

(kwK)

up(pk+1w k+1w gD) +

wt

up he

(pk+1w poutw ), z

e.

The basic overall discrete system is given by (16) and (20). Below wediscuss two important modifications.

3.2 Projection to H(div)-space

In this section we show an alternative way of handling the flux needed inthe pressure step (17) when l = 1. The idea is to project the flux into theH(div, )-space. In practice, we project the flux onto the space defined bythe lowest order Raviart-Thomas elements(RT0). For more details on mixedelements see, e.g., [3].

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We look for the flux in the space spanned by RT0-elements

V={v H(div, ) :v|ERT0(E) , E Th}. (21)

Using the divergence theorem, v, 1)E=v , 1E (22)

and the fact that for RT0 elements the divergence inside the elements isconstant, as are the normal fluxes over each edge, gives

u|E=

1

|E|

u , 1E (23)

for allu V. Using equation (19) we get uk+1w |E

= 1

|E|

(kwK)

up{{(pk+1w k+1w gD) }} +

wt

up he

[[pk+1w ]], 1E

.

(24)

For edges on the no-flow boundary, the integral is zero. For edges on eitherthe inflow or outflow boundaries, we do not use the mean value, and the

jump is against the given boundary data, similar to equation (20). After

this local, intermediate step between the pressure and saturation steps, wenow have divergence of the wetting phase flux in each element. Therefore,the saturation step becomes: FindSk+1w W

m, m >0, such that

ETh

k+1w S

k+1w

t , z

E

=ETh

kwSkwt

, zE

+

fw, zE

uk+1w , zE (25)for all z Wm. This is our alternative to (20).

Remark 3.1. If we use piecewise constant saturations, Sw W0

, we willnaturally project the fluxes to RT0. This is because on Ethe test functionz = 1, and hencez= 0in equation (20). Meaning that the alteration (25)is equivalent to the original formulation (20).

Remark 3.2. Suppose the saturationSkw is piecewise constant, and that thedata is piecewise constant. Then using projected fluxes (25) will give piece-wise constantSk+1w , even ifSw W

m,m 1. This is because the saturationstep is, in this particular case, just an L2-projection onto piecewise constants.

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3.3 Penalty for discontinuity in capillary pressure

In the formulation above, the effect of capillary pressure appears explicitlyonly in the pressure step (16). In the saturation step, (20) or (25), the capil-lary effect appears implicitly through the flux that depends on the pressure.We propose that the capillary effect is taken into account explicitly also inthe saturation step. The idea is to add a penalty to discontinuity of the cap-illary pressure. This is a valid addition, since the capillary pressure shouldbe continuous. Recall that the saturation is not necessarily continuous ifthe capillary pressure curves are different.

For simplicity we will work with saturation step (25) but the exact sameprocedure works for (20), as well. We propose to add to (25) the term

eGint

h

(1 + {{pkc}}) t

[[pk+1c ]], [[z]]e, (26)

in which > 0 is a penalty parameter. The negative sign comes from thefact that capillary pressure is often a decreasing function of wetting phasesaturation, andz is a test function for wetting phase saturation. Hence thisterm is positive, which corresponds to a sign convention chosen in equations(25) and (20). The scaling 1/

1 + {{pkc}}

ensures that the penalty term is

of the same scale as the rest of the terms. There is no need to scale withmesh size he, since using inverse trace inequalities one can bound this termwith the H1()-norm of the capillary pressure and the L2()-norm of the

saturation, which correspond to the correct solution spaces. The factor1 inthe denominator is to prevent division by 0 when the capillary pressure issmall.

Unfortunately pkc = pc(Skw) is a non-linear function of saturation. To

maintain computational efficiency, we linearize this term. Using a linearapproximation

pk+1c =pc(Sk+1w ) pc(S

kw)+p

c(S

kw) (S

k+1w S

kw) =p

kc +p

kc (S

k+1w S

kw), (27)

we get

eGint

h

(1 + {{pkc}}) t[[pkc Sk+1w ]], [[z]]e

=

eGinth

(1 + {{pkc}}) t[[pkc S

kwp

kc ]], [[z]]e, (28)

which is a linear with respect to the unknown Sk+1w . With this penalty forcapillary pressure in place of (26), and using the H(div, )-projection, the

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saturation step reads

ETh

k+1w S

k+1w

t , z

E

eGinth

(1 + {{pkc}}) t[[pkc S

k+1w ]], [[z]]e

=ETh

kwSkwt

, zE

+

fw, zE

uk+1w , zE

eGint

h

(1 + {{pkc}}) t[[pkc S

kwp

kc ]], [[z]]e. (29)

Remark3.3. Suppose there is only one capillary pressure curve for the wholedomain. If the curve is constant, the added terms will vanish, which is

exactly what we want. If the curve is monotone, the added terms willenforce the continuity of saturation, which is the correct physical behaviorfor monotone capillary pressure.

To close this section, the overall discrete system is given by (16) and(29) with (24) when one uses the alternativeH(div) velocityuk+1w . Iterativecoupling as described in section 2.3 can also be applied to the system.

4 Numerical examples

In this section we discuss numerical solutions to the slightly compressible

two-phase flow problem using the DG-DG discretization described in section3. Except where noted, we choose the pressure and saturation of the wettingphase, Pw and Sw, as the primary variables.

The simulations are run for 600days, starting with a time step of0.001days and which increases over time by the multiplier1.2to a maximum timestep of0.5days. As mentioned earlier it is useful to take shorter time steps inthe saturation equation, so we take 10 saturation steps for every 1 pressurestep. The computational domain is (x,y ,z) [0, 500] [0, 500] [100, 0].As shown in figure 1, the inflow boundary, inis (x,y ,z) {0} [400, 500] [100, 0], and the outflow boundary, out is (x,y ,z) {500} [0, 100][100, 0]. The rest of the boundary,0is of no flow type. The initial pressure

is set to 500psi and the pressure boundary conditions on the inflow andoutflow boundaries are550psi and450psi respectively. The initial saturationis 0.2 and, in order to imitate a water flood, the inflow boundary conditionis set to 1.

To make sure there is no instability coming from the pressure equation,we use the non-symmetric interior penalty Galerkin (NIPG) method (= 1in (16)). In all our numerical examples we use a piecewise linear basis, l = 1,to approximate the pressure. A pressure penalty of10 is used, theoreticallyany penalty greater than 0 is acceptable and numerical tests show that thesolution is not sensitive to this parameter when it ranges from 5 to 50.

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Figure 1: Computational domain as seen from above.

0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

30

Capillary Pressure

Water Saturation

CapillaryPressure

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.91

Relative Permeability Curves

Water Saturation

RelativePermeability

Oil

Water

Figure 2: Capillary pressure for validation example on the left. Relativepermeabilities for all the examples on the right.

Except where noted, we use linear approximation for the saturation,m = 1, and the penalty for the jump in capillary pressure. The saturationpenalty parameter is set to 10. The numerical solutions are not sensitiveto this parameter when it is between 1 and 100. In addition, we use theprojection to H(div)-space except where noted.

The capillary pressure and relative permeability curves are similar to[9]. These capillary pressure curves are of the formpc(Sw) =

10K

log(Sw),

i.e., inversely proportional to the permeability. Furthermore, the relativepermeabilities are given as a function of the wetting phase saturation askw =S

2w andkn = (1 Sw)

2, see figure 2.

4.1 Validating the method

In this section we consider the problem where there is only one capillarypressure curve and take K= 100 millidarcy, see figure 2.

In order to determine the solution to the problem we model the satu-ration with piecewise constant functions, m = 0, and view the solution on

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Figure 3: Validation Example: Piecewise linear pressure solutions on theleft. Piecewise constant saturation solutions on the right

different size meshes. For these numerical tests the penalty for the discon-tinuity in capillary pressure (28) was not used. In addition, the projectionto the H(div)-space (24) does not change the previous formulation (see re-mark 3.1). The numerical results are shown in figure 3. Notice that as werefine the mesh, the solution remains the same. For these reasons we takethese results to be a valid approximation of the solution that we can use asa baseline with which to compare further results. Henceforth we will notinclude the pressure curves, but we note that they are all consistent withthose shown here.

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Figure 4: Validation example: Piecewise linear saturation solutions, on theleft using no slope limiter, on the right using a global slope limiter.

Using piecewise linear functions to approximate the saturation, m = 1,we obtain the results in figure 4. These results include the penalty for dis-continuous capillary pressure and the penalty parameter is set to 10. Thenumerical solutions are not sensitive to this parameter when it is between1 and 100. There is some grid dependance on the coarser mesh, but thisvanishes as the mesh is refined. The flux projection removes oscillations, sothere is no need to use a local slope limiter. However, there are overshoots atthe inflow, which increase over time. On the fine grid, the maximum valueof the saturation is 1.163 which occurs at day 600. To handle this problem,

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Figure 5: Validation example: Illustration of remark 3.2. Piecewise constant

saturation on the left and piecewise linear saturation on the right.

we implemented a global slope limiter that enforces the saturation to staybetween its maximum value 1 and minimum value 0. We use a limiter thatguarantees that the saturation stays bounded without any tuning parame-ters. This limiter is discussed in [14]. We further note that the saturationplume travels the same distance using the piecewise linear basis functionsas it did when we used the piecewise constants. This validates the proposedimprovements, that is, the projection to H(div) and adding a penalty forthe jump in the capillary pressure.

It was previously remarked (see remark 3.2) that if at some time tk the

saturation Skw is piecewise constant, then using projected fluxes will give apiecewise constant solution, Sk+1w , at timetk+1. Figure 5 shows when thereis no penalty to the discontinuity of capillary pressure, the saturation at600 days is the same regardless if your test functions are piecewise linearor piecewise constant. Observe also that here the projection to theH(div)space removes the local oscillations from the (in principle) piecewise linearsaturation.

4.2 Two Rock Types

In this section we look further at the improvements to the method by consid-

ering the case where there are two rock types. In the results in this sectionwe are using the same computational domain as in the previous section,except now there is a second rock type located in the domain in the area(x,y ,z) [0, 300] [200, 300] [100, 0], as shown in figure 6.

The rocks will have different capillary pressure curves due to the differ-ence in the permeabilities of the rocks. Rock type 1 has a permeability of100 millidarcy, and the second rock type has a permeability of 10 millidarcy.The capillary curves are shown in figure 6.

This time we take the initial condition for the saturation to be piecewiseconstant with a value ofSw = 0.2in rock type 1, and in rock type 2 it is set

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0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

30

Water Saturation

CapillaryPressure

Capillary Pressure

Rock 1

Rock 2

Figure 6: Two rock types example: Location of the two rocks in computa-

tional domain and the associated capillary pressure curves.

to satisfy that the capillary pressure is continuous across the boundary.We used the global slope limiter with these results and set the global

max and min to be 1 and 0.2 respectively.Shown in figure 7 are the saturation and pressure results at various

times. We note that the pressure solution is smooth as we would expect.Looking at the saturation at the boundary of the rock types, we notice thatthe saturation is discontinuous, as expected. For example, in rock type 2,the saturation is pooling up before it can seep into rock 1 so that it willrespect the continuity of the capillary pressure that is explicitly required inthe saturation step. We also see that because of the higher permeability inrock 1, the saturation front moves faster around rock 2 than going throughit. And by 1000 days the flow has come around the bend of the second rocktype and is heading towards the outflow boundary.

4.3 Choosing the primary pressure

In this section we illustrate the problems that can occur with using thewetting phase as the primary pressure. We look at the same example oftwo rock types as before, except we change the initial condition so thatthe Sw = 0 everywhere, which is a valid choice with respect to capillary

pressure curves in figure 6. We also use the global slope limiter to enforcethe global max and min value to be 1 and 0 respectively. When we do this,we get the results in figure 8. This simulation shows large oscillations in boththe pressure and saturation solutions. The oscillations in the pressure aredifficult to see, but they are the main source of oscillations to the saturationsolution, which is evident in the figures. Observe that the oscillations areworse in rock 1 where the capillary pressure has a steeper gradient near theresidual saturation, which supports the discussion on choosing the referencepressure in section 2.3.

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PWAT

550

540

530

520

510

500

490

480

470

460

450

SWAT

1

0.95

0.9

0.85

0.8

0.75

0.7

0.65

0.6

0.55

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Figure 7: Two rock types example: Linear water pressure and saturation inthe example with two rock types. From top to bottom at day 300, 600, and1000.

We ran the problem with the same parameters but with the non-wettingphase as the primary pressure. Due to a computer code limitation thischanges our pressure boundary condition to be with respect to the non-wetting phase. Thus, this is not exactly the same problem as in the previouscase, but it is very similar. We choose to look at times where the saturationfront had reached approximately the same distance as in the previous case tocompare the results, see figure 9. The speed of the front is different due to theobviously incorrect results obtained when using the water pressure. Observe

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PWAT

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Figure 8: Primary pressure choice example: Water pressure and water satu-ration with initial saturation 0 and using water pressure as primary pressure.From top to bottom at day 500, 1000, and 1500.

that the speed of the front is similar to figure 7. The main improvement isof course that using the the non-wetting phase pressure, we no longer haveoscillations in the solution. The reason for the improvement is that we areavoiding the numerical instability problem discussed in section 2.3.

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POIL

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Figure 9: Primary pressure choice example: Oil pressure and water satu-ration with initial saturation 0 and using oil pressure as primary pressure.From top to bottom at day 300, 700, and 1000.

5 Conclusions

In this work we proposed two improvements to the discontinuous Galerkinformulation of the slightly compressible two phase flow solved using IMPESscheme, namely projecting the velocity to the H(div) space and includingan explicit penalty for the jump in the capillary pressure in the saturationequation. We showed that projecting the velocity to the H(div) space re-duces the local oscillations to the limit where we no longer need local slope

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limiters. However, we did observe a need for a global slope limiter. The

global slope limiter is much easier to implement and does not affect thedynamics of the solution as severely as a local limiter.

We showed that adding the penalty for the jump in the capillary pres-sure improved the saturation step considerably. With this formulation weobserved that the saturation always fulfilled the continuity of the capillarypressure even if the magnitude of the capillary pressure was small comparedto the phase pressures. In future work we will examine the use of this penaltyfor the capillary pressure in mixed formulations, e.g., in the multipoint fluxmixed finite element (MFMFE) method.

Lastly we showed that choosing the non-wetting phase pressure as theprimary pressure is more stable numerically if the capillary pressure has

steep gradients, as it usually has near the wetting phase residual saturation.

References

[1] Center for Subsurface Modeling. http://csm.ices.utexas.edu/.

[2] Todd Arbogast, Gergina Pencheva, Mary F. Wheeler, and Ivan Yotov.A multiscale mortar mixed finite element method. Multiscale Model.Simul., 6(1):319346 (electronic), 2007.

[3] Franco Brezzi and Michel Fortin. Mixed and hybrid finite element

methods, volume 15 ofSpringer Series in Computational Mathematics.Springer-Verlag, New York, 1991.

[4] Zhangxin Chen, Guanren Huan, and Yuanle Ma. Computational meth-ods for multiphase flows in porous media. Computational Science &Engineering. Society for Industrial and Applied Mathematics (SIAM),Philadelphia, PA, 2006.

[5] Holger Class, Anozie Ebigbo, Rainer Helmig, Helge Dahle, Jan Nord-botten, Michael Celia, Pascal Audigane, Melanie Darcis, JonathanEnnis-King, Yaqing Fan, Bernd Flemisch, Sarah Gasda, Min Jin, Ste-fanie Krug, Diane Labregere, Ali Naderi Beni, Rajesh Pawar, Adil Sbai,

Sunil Thomas, Laurent Trenty, and Lingli Wei. A benchmark study onproblems related to co2 storage in geologic formations. ComputationalGeosciences, 13(4):409434, 2009.

[6] Clint Dawson, Shuyu Sun, and Mary F. Wheeler. Compatible algo-rithms for coupled flow and transport. Comput. Methods Appl. Mech.Engrg., 193(23-26):25652580, 2004.

[7] Y. Epshteyn and B. Rivire. Fully implicit discontinuous finite elementmethods for two-phase flow. Appl. Numer. Math., 57(4):383401, 2007.

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[8] A. Ern, I. Mozolevski, and L. Schuh. Discontinuous galerkin approxi-

mation of two-phase flows in heterogeneous porous media with discon-tinuous capillary pressures. Computer Methods in Applied Mechanicsand Engineering, 199(23-24):1491 1501, 2010.

[9] Hussein Hoteit and Abbas Firoozabadi. Numerical modeling of two-phase flow in heterogeneous permeable media with different capillaritypressures. Advances in Water Resources, 31(1):5673, 2008.

[10] Donald W Peaceman. Fundamentals of numerical reservoir simulation,volume 6. Elsevier Scientific Pub. Co. : distributors for the U.S. andCanada, Elsevier North-Holland, Amsterdam, 1977.

[11] Shuyu Sun and Mary F. Wheeler. Symmetric and nonsymmetric dis-continuous Galerkin methods for reactive transport in porous media.SIAM J. Numer. Anal., 43(1):195219, 2005.

[12] Shuyu Sun and Mary F. Wheeler. Projections of velocity data for thecompatibility with transport. Comput. Methods Appl. Mech. Engrg.,195(7-8):653673, 2006.

[13] Mary F. Wheeler, Guangrie Xue, and Ivan Yotov. A multipoint fluxmixed finite element method on distorted quadrilaterals and hexahedra.ICES report, 10-34, August 2010.

[14] Xiangxiong Zhang and Chi-Wang Shu. On maximum-principle-satisfying high order schemes for scalar conservation laws. Journal ofComputational Physics, 229(9):3091 3120, 2010.

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