Bonecos.pdf

8/11/2019 Bonecos.pdf

1/16

Adaptive simulations of two-phase flow by discontinuousGalerkin methods

W. Klieber, B. Riviere *

Department of Mathematics, University of Pittsburgh, 301 Thackeray, Pittsburgh, PA 15260, USA

Received 4 October 2005; received in revised form 12 April 2006; accepted 3 May 2006

Abstract

In this paper we present and compare primal discontinuous Galerkin formulations of the two-phase flow equations. The wettingphase pressure and saturation equations are decoupled and solved sequentially. Proposed adaptivity in space and time techniques yieldaccurate and efficient solutions. Slope limiters valid on non-conforming meshes are also presented. Numerical examples of homogeneousand heterogeneous media are considered. 2006 Elsevier B.V. All rights reserved.

Keywords: Error indicators; Discontinuous Galerkin; Adaptive time stepping; Slope limiters; Five-spot; NIPG; SIPG; IIPG; OBB

1. Introduction

Accurate simulations of multiphase processes are essen-tial in problems related to the environment and the energy.There is a need for discretization methods that performwell on very general unstructured grids. Standard methodssuch as the finite difference methods, finite volumes andexpanded mixed finite element fail to capture the flow phe-nomena in the case of highly heterogeneous media withfull permeability tensors. Recently, discontinuous Galerkin(DG) methods have been applied to a variety of flow andtransport problem[21,22,2,25,24]and due to their flexibil-ity, they have been shown to be competitive to standard

methods. Furthermore, DG methods allow for unstruc-tured meshes and full tensor coefficients. Even though thediscontinuous finite element methods are more expensivethan the finite difference methods, oil engineers are willingto pay the price for accuracy and thus avoid costly mis-takes[20].

In this work, the pressure-saturation formulation (alsoknown as the sequential formulation) of the two-phase flow

problem is discretized using several discontinuous Galerkin

methods. Description of the sequential model and otherformulations for two-phase flow can be found in [14,6].The unknowns are the wetting phase pressure and satura-tion and the equations are solved successively. One imme-diate advantage is the fact that the difficulty arising fromthe non-linearity is removed by time-lagging the co-efficients. The pressure equation is solved by the OdenBaumannBabuska (OBB) method [18] whereas thesaturation equation is solved by either the OBB, the non-symmetric interior penalty Galerkin method (NIPG) [23],the symmetric interior penalty Galerkin method (SIPG)[27,1] or the incomplete interior penalty Galerkin method

(IIPG) [9,26]. One can note that all four methods OBB,NIPG, SIPG and IIPG are very similar to each other,and can be described by the same variational formulationwith a bilinear form involving constant parameters. Forinstance, OBB and NIPG only differ by the addition of apenalty term; SIPG and NIPG only differ by a sign.

The objective of this work is to investigate adaptivesimulations in time and space on unstructured meshes.We formulate error indicators for the spatial refinementand derefinement techniques. We also present an algorithmthat allows the time step to vary during the simulation. One

0045-7825/$ - see front matter 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.cma.2006.05.007

* Corresponding author. Tel.: +1 412 6248315; fax: +1 412 6248397.E-mail address:[email protected](B. Riviere).

www.elsevier.com/locate/cma

Comput. Methods Appl. Mech. Engrg. 196 (2006) 404419
mailto:[email protected]:[email protected]


2/16

of the main difficulties is the development of slope limitersthat would handle meshes with hanging nodes. We proposea limiting technique based on the one introduced byDurlofsky et al. [10] for conforming meshes. By the useof adaptivity, we significantly reduce the computationalcost while keeping the accuracy. To our knowledge, there

is little work in the literature on applications of DG meth-ods to two-phase flow. In [19,4], simulations were per-formed on uniformly refined meshes and with a constanttime step. In[3], DG is applied to a total pressure-satura-tion formulation. In [17], DG and mixed finite elementsare coupled. More recently, in [13], a compressible airwater two-phase flow problem is numerically solved onuniform meshes using the NIPG/OBB/SIPG method anda local discontinuous Galerkin (LDG) [8] discretizationfor the saturation equation. In this case, a Kirchoff trans-formation is required to obtain a diffusive flux from theprevious time step. The saturation equation is solvedexplicitly in time, which is computationally appealing;

however this reduced cost is compensated by the introduc-tion of an additional unknown, intrinsic to the LDG for-mulation. Finally, in [11], fully coupled DG formulationsare considered and in this case slope limiters are not neededeven for high order of approximation. However, the solu-tion of the fully coupled DG formulations require the con-struction of a Jacobian matrix at each time step for theNewtonRaphson method.

The plan of the paper is as follows. In the next section,we present the equations describing the two-phase flowproblem. Section3contains the discrete schemes and nota-tion. The adaptive strategy in space and time, as well as

the slope limiting technique, are described in Section 4.Numerical examples are given in Section 5. Some conclu-sions follow.

2. Model problem

The mathematical formulation of two-phase flow ina porous medium X in R2 consists of a coupled systemof non-linear partial differential equations. The phases con-sidered here are a wetting phase (such as water) and anon-wetting phase (such as oil). For each phase, the conser-vation of mass and a generalized Darcys law are obtained.Under the assumption of incompressibility, a pressure-saturation formulation is derived, for which the primaryvariables are the pressure and the saturation of the wettingphase denoted by pw and sw:

r ktKrpw r koKrpc; 1o/sw

ot r kokw

ktKrpc

r kw

ktut

: 2

The coefficients in Eqs. (1) and (2) are defined below:

Kis the permeability tensor and is spatially dependent;for heterogeneous media, K is discontinuous.

The coefficient / denotes the porosity of the medium.

kt= ko+ kwis the total mobility, that is, the sum of themobility of the non-wetting phase and the mobility ofthe wetting phase. Mobilities are functions that dependon the fluid viscosities lw and lo and on the effectivewetting phase saturation se. The effective saturationdepends on the residual wetting phase and non-wetting

phase saturations srw and sro as follows:se swsrw

1:0srwsro :

The mobilities are then given by the BrooksCoreymodel[5]:

kwsw 1lw

s4e ; kosw 1

lo1se21s2e:

The difference of the pressures of the two phasespc= pnpw is the capillary pressure. From theBrooksCorey model, it depends on the effective satura-tion and a constant entry pressure pd:

pcsw pdffiffiffiffisep :From this equation, we see that p0csw< 0 and we willwrite:rpc jp0cjrsw.

ut= uo+ uw is the total velocity, that is the sum of thetwo phases velocities. Each phase velocity is given as

ud Kkdrpd; do; w:Let n denote the outward normal to oX. We associate toEqs. (1) and (2) several boundary conditions, by firstdecomposing the boundary of the porous medium oXinto disjoint parts:

oXCp1[Cp2Cs1[Cs2; Cp1\Cp2Cs1\Cs2 ;:

The boundary conditions for (1) are of Dirichlet andNeumann type:

pwpdir; on Cp1; 3Kktrpw n0; on Cp2: 4The boundary conditions for (2) are of Robin andNeumann type:

swut

Kkokw

kt

p0crsw nsinut n; on Cs1; 5

Kkokwkt

p0crsw

n0; on Cs2: 6

3. Scheme

In this section, we first establish some notation for thetemporal and spatial discretization and we present ournumerical scheme. Let 0 = t0


3/16

The domain X is subdivided into triangular elementsthat form a mesh. Because of the refinements and derefine-ments, the mesh changes at every time step. Let us denoteby Eih fEgEthe mesh at timeti+1. Lethi be the maximumdiameter of the elements. Let Chi be the union of the opensets that coincide with interior edges of elements ofEih. Let

edenote a segment ofChi shared by two trianglesEk

andEl

ofEih (k>l); we associate with e, once and for all, a unitnormal vector nedirected from E

k to El and we define for-mally the jump and average of a function w on e by

w wjEkje wjElje; fwg 1

2wjEkje

1

2wjElje:

Ife is adjacent to oX, then the jump and the average ofwonecoincide with the trace ofwoneand the normal vectorne coincides with the outward normal n. The quantityjejdenotes the length ofe.

For each integerr, we define a finite element subspace ofdiscontinuous piecewise polynomials:

DrEih fv : vjE2 PrE 8E2 Eihg;where Pr(E) is a discrete space containing the set of poly-nomials of total degree less than or equal tor on E. We willapproximate the wetting phase pressure and saturation bydiscontinuous polynomials of order rpand rs, respectively.

We now derive the variational formulation for the two-phase flow problem, by considering the pressure Eq. (1)and the saturation Eq.(2) separately.

3.1. The pressure equation

We rewrite(1) by defining v Kkorpc Kkojp0cjrsw:r Kktrpw r v: 7Multiplying (7) by a test function v2Drp , and using

Greens formula on one element Eyields:

b1

bb

E2

23

E1

E3

E4E0

Fig. 1. Slope limiting on non-conforming meshes.

Fig. 2. Refinement of a triangular element.

Fig. 3. Five-well example: coarse mesh at initial time and adaptive meshes obtained at 15 and 45 days.

406 W. Klieber, B. Riviere / Comput. Methods Appl. Mech. Engrg. 196 (2006) 404419
http://-/?-http://-/?-


4/16

ZE

Kktrpw rvZ

oE

KktrpwnEv

ZE

v rvZ

oE

vnEv;

where nEis the outward normal to E. Summing over all theelements in Ei

h

and using the fact that pwand v are smoothenough, namely [pw] = 0, [Kkt$pw ne] = 0 and [v ne] = 0,we haveXE2Ei

h

ZE

Kktrpw rvX

e2Cih[oX

Ze

fKktrpwnegv

Xe2Ci

h

Ze

fKktrvnegpw

XE2Ei

h

ZE

v rvX

e2Ch[oX

Ze

v nev:

Making use of the boundary conditions (3) and (4), weobtain

XE2Ei

h

ZE

Kktrpw rvX

e2Cih[Cp1

Ze

fKktrpwnegv

X

e2Cih[Cp1

Ze

fKktrv negpw

XE2Ei

h

ZE v rv Xe2Cih[oX Ze v nev

Xe2Cp1

Ze

Kktrv npdir: 8

3.2. The saturation equation

Similarly, we define the auxiliary vector f kwktut. Then,

(2)can be rewritten as

o/swot

r Kkokwkt

jp0cjrsw

r f: 9

As for the pressure equation, we multiply by a test functionz2Drs over one element in Eih, sum over all elements, and

Fig. 4. Five-well example: three-dimensional pressure contours at 15, 30, 45 and 52.5 days.

W. Klieber, B. Riviere / Comput. Methods Appl. Mech. Engrg. 196 (2006) 404419 407
http://-/?-http://-/?-


5/16

use the regularity ofswand f. We finally obtain after somealgebraic manipulation:ZX

o/swot

zXE2Ei

h

ZE

Kkokw

ktjp0cjrsw rz

Xe2Cih[oX Ze K

kokw

kt jp0

cjrs

wne z

Xe2Ci

h

Ze

Kkokw

ktjp0cjrz ne

sw

XE2Ei

h

ZE

f rzX

e2Cih[oX

Ze

fnez:

Making use of the boundary conditions (5), (6) and thecontinuity of pressure, we have:ZX

o/swot

z

XE2EihZE

Kkokw

ktjp0cjrsw rz

Xe2Cs1

Ze

swut nezXe2Ci

h

Ze

Kkokw

ktjp0cjrsw ne

z

Xe2Ci

h

Ze

Kkokw

ktjp0cjrz ne

sw

Xe2Ci

h

r

jejZe

swz

XE2Ei

h

ZE

f rzX

e2Cih[oX

Ze

fnez Xe2Cs1

Ze

sinutnez

Xe2Ci

h

Ze

kw

ktfKktrz negpw

XCp1

ZeKkwrz nepwpdir: 10

The equation above is parametrized by the coefficients2{1,0,1} and rP 0. For a positive penalty value r,the choice =1 yields the SIPG method, the choice= 0 yields the IIPG method and the choice = 1 theNIPG method. If r= 0 and = 1, we obtain the OBBmethod.

3.3. The discrete scheme

We discretize the time derivative by finite difference,

which yields the backward Euler scheme. The initialapproximationsP0w, S0ware simply obtained by aL

2 projec-tion of the initial datapw(t= 0) andsw(t= 0). Based on(8)and (10), we formulate the following numerical method:

given Piw; Siw 2DrpDrs , find Pi1w ; Si1w 2DrpDrssuch that for allv;z 2DrpDrs :

Fig. 5. Five-well example: three-dimensional saturation contours at 15, 30, 45 and 52.5 days.



6/16

XE2Ei

h

ZE

KktSiwrPi1w rv Xe2Ci

h[Cp1

Ze

fKktSiwrPi1w negv

X

e2Cih[Cp1

Ze

fKktSiwrv negPi1w

XE2Ei

h

ZE

vih rvX

e2Cih[oX

Ze

vi"h nev

Xe2Cp1

Ze

KktSiwrvnpdir 11

andZX

/

DtiSi1w z X

E2Eih

ZE

KkoSiwkwSiw

ktSiw jp0cSiwjrSi1w rz

Xe2Cs1

Ze

Si1w Uitnez

Xe2Ci

h

Ze

KkoSiwkwSiw

ktSiw jp0cSiwjrSi1w ne

z

Xe2Ci

h

Ze

KkoSiwkwSiw

ktSiw jp0cSiwjrzne

Si1w

Xe2Cihr

jejZe

Siwz

ZX

/

DtiSiwz

XE2Ei

h

ZE

fih rz

X

e2Cih[Cp1

Ze

fi"h nez

Xe2Cs1

Ze

sinUit ne

Fig. 7. Two-dimensional pressure contours at 7.5, 15, 22.5 and 30 days.

INJECTION

PRODUCTION

Fig. 6. Domain and coarse mesh for quarter-five spot.



7/16

Xe2Ci

h

Ze

kwSiwktSiw

fKktSiwrz negPiw

Xe2Cp1

Ze

KkwSiwrznePiwpdir; 12

where Ui

t

, fih

and v ih

are the approximates of uit

; fi and vi.

Uit KkwSiwrPiwKkoSiwp0cSiwrSiw rPiw;

vih KkoSiwjp0cSiwjrSiw;

fih

kwSiwktSiw

fUitg:Because of the discontinuous approximations, there aretwo values for the functions vi and fi on an interior edge.These quantities are then replaced by the upwind numericalfluxes vi"h and f

i"h . Upwinding is done with respect to the

normal component of the average of the total velocity Ut

8e

oEk

\oEl;

k> l

;

8w; w"

wjEk iffUitg ne P 0;wjEl iffUitg ne < 0:(

From the derivations in Sections3.1 and 3.2, we obtain theconsistency of the scheme(11) and (12).

Lemma 1. If (pw, sw) is a solution of(1), (2), then (pw, sw) isalso a solution of(11) and (12).

3.4. Local mass balance

Let us fix an element Eand a test function v2Drp thatvanishes outside ofE. For simplicity, we assume that E isan interior element in X. The pressure equation (11)becomes:

ZE

KktSiwrPi1w rv ZoE

fKktSiwrPi1w nEgv

Z

oE

1

2KktSiwrvnEPi1w

ZE

vih rvZ

oE

vi"h nEv:

If in addition, we let v to be equal to one overE, we obtainthe local mass property satisfied by the approximations:

Z

oE

fKktSiwrPi1w nEgvZ

oE

vi"h nEv0:

3.5. Slope limiting

Approximations of high order yield overshoot andundershoot in the neighborhood of the front of the injectedphase. Slope limiters are the appropriate tools for decreas-ing the local oscillations[7,15]. To our knowledge there is

Fig. 8. Two-dimensional saturation contours at 7.5, 15, 22.5 and 30 days.



8/16

no analysis available for slope limiters in 2D and 3D, evenon a conforming mesh. We are also not aware of limitersthat would handle non-conforming meshes. In this section,we propose a limiting technique that can handle mesheswith hanging nodes. This procedure is successfully testedfor our two-phase flow problem. We apply the limiting

technique to the approximations Pi

1

w and Si

1

w after eachtime step ti+1.In what follows, we say that one element Eis active if it

belongs to the mesh Eih, i.e., if it is used in the computationof (11) and (12). The element can become inactive if it isrefined and thus its children are created and activated.The limiting process consists of two steps.

First, we loop through all the active elements startingfrom the oldest generation to the youngest (in general thiswould mean that the order is in decreasing size). For exam-ple, Fig. 1 shows an example of five elements of differentgeneration: ifGdenotes the generation of the elements E0andE1, then elementsE3and E4are of younger generation

G+ 1 and elementE2is of older generation G1. Thus, itis assumed that the limiting process has been alreadyapplied to E2.

(1)Neighbor averages: We first compute the average sat-uration for the element to be limited and all neighboringelements as follows. Let S0 denote the average saturation

overE0and letSjdenote a function associated to each sidej2{1,2,3} of E0. For E0 and the neighbors of the samegeneration, we have the usual averaging operator:

S0AE0; S1AE1; where AE 1jEjZE

Si1w :

To compute S2 corresponding to the side 2 ofE0 and theelement E2 that is of older generation, we first locate thebarycenter b2 of an imaginary childeC2 of the same gener-ation ofE0 (see dashed lines inFig. 1). We then set

S2Si1w jE2b2:We note that S2AeC2. The smaller elements E3and E4belong to a parenteE(see dotted lines in Fig. 1). If we de-note by F

~E1 ;. . . ;F

~E4 the children of

eE, we can writeS31

4

X4l1BF~El;

where the function B is defined recursively as (using thenotation FEl for the lth child ofE):

BE

1

jEjZE

Siw; if Eactive;

1

4

X4l1BFEl; otherwise:

8>>>>>:

Fig. 9. Two-dimensional pressure contours at 7.5, 15, 22.5 and 30 days obtained on uniformly refined meshes.



9/16

If the edgejis a boundary edge, then Sj is defined accord-ing to the boundary conditions.

Sjsin on Cs1; SjS0 on Cs2:(2) Test: We then compute the saturation Si1w jE0mj

evaluated at the midpoint mjof each edge jand we check

that this value is betweenSjand S0. We stop here if the testis successful, otherwise we continue to step 3.(3)Construction of three linears: Based on the technique

by Durlofsky et al.[10], we construct three linears using thepoints bjand the averages Sj. For instance, if we write thelinears as Ljx;y aj0aj1xaj2y, for j2{1,2,3}, theyare uniquely determined by

Ljb0 S0 and Ljbl Sl; for l6j:We then rank the linears by decreasing

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaj12 aj22

q and

check that for the values of the linears evaluated at themidpoint ml, Ljml, is between Sl and S0 for 1P lP 3.If none of the constructed linears satisfy the test then the

slope is reduced to 0.Second, we loop through all elements and check that

their slopes are not too large in the euclidean norm. If itis larger than a cut-off value (set up by user), we scale itby the ratio cut-off/norm.

4. Adaptivity strategy

In this section, we define the error indicators and presentthe adaptivity in space and time techniques. They are basedon the a posteriori error estimates obtained for a linearconvectiondiffusion time-dependent problem [12], that

has some similarity with the saturation equation. However,there is no rigorous mathematical proof for our coupledsystem of equations and the error estimators of [12] areused here as error indicators in the adaptivity algorithm.

4.1. Error indicators

We define the following quantities:

Rvol /Dti

Si1w Siwr KknSiwkwSiw

ktSiw jp0cSiwjrSiw

r kwS

iw

ktSi

wU

it ;

Re1 Si1w ;

Re2 KkwSiwknSiwktSiw

jp0cSiwjrSi1w n

fUitg nSi1w ;

Fig. 10. Two-dimensional saturation contours at 7.5, 15, 22.5 and 30 days obtained on uniformly refined meshes.



10/16

Res1

sinU

it

n

Si1w U

it

n

KknSiwkwSiw

ktSi

w jp0c

Siw

jrSi1w

n;

Res2KknSiwkwSiwktSiw

jp0cSiwjrSi1w n:

Then the error indicator gEcomputed on each element Eis

gE h4EkRvolk20;EX

e2oEnoXh3ekRe2k20;e he1kRe1k20;e

X

e2oE\Cs1h3ekR es1k20;e

Xe2oE\Cs2

h3ekR es2k20;e!1=2

;

X

Sw

0 100 200 300 4000

0.2

0.4

0.6

0.8

1

X

Pw

0 100 200 300 4002.4E+06

2.6E+06

2.8E+06

3E+06

3.2E+06

3.4E+06

Fig. 11. Saturation (left) and pressure (right) fronts along the diagonal line x =y at 7.5, 15, 22.5 and 30 days. The solid line corresponds to a uniformmesh refinement (h3) and the dashed line to an adaptively refined mesh.

X

Sw

0 100 200 300 4000

0.2

0.4

0.6

0.8

1

X

Pw

0 100 200 300 4002.4E+06

2.6E+06

2.8E+06

3E+06

3.2E+06

3.4E+06

Fig. 12. Saturation (left) and pressure (right) fronts along the diagonal linex=yat 15 and 30 days. Solid lines correspond to methods NIPG, SIPG, IIPG

with r = 10

5 and OBB. Dashed lines correspond to methods NIPG, SIPG and IIPG with r = 1.

Table 1Number of degrees of freedom for adaptive and non-adaptive simulationsin the case (,r) = (1,0)

t(days) DOFS press DOFS sat

AMR UNI AMR UNI

7.5 2538 25,344 1269 12,67215 2412 25,344 1206 12,67222.5 2142 25,344 1071 12,67230 2466 25,344 1233 12,672 Table 2

Total number of degrees of freedom for adaptive simulations for allmethods

t(days) OBB NIPG NIPG SIPG SIPG IIPG IIPG

r= 0 r= 1 r= 10

5

r= 1 r= 10

5

r= 1 r= 10

5

7.5 3807 3348 3159 3294 3294 3348 402315 3618 3105 2970 3132 2997 3105 359122.5 3213 3348 3213 3483 3267 3375 337530 3699 2538 3780 2484 3834 2997 3915



11/16

where hE is the diameter of the element E and hemaxhEk; hEl if the edge e is shared by elements Ek andEl. Note that the notation oEnoX means that the edgesare interior edges only.

4.2. Adaptive mesh refinement technique

Let us assume that the solution Piw and Siw have been

obtained at the ith time step. We compute the error indica-tor gEfor each active element E. Then, we first refine theappropriate elements and second apply the derefinementtechnique.

Refinement: We refine each element whose error indica-tor is greater than a threshold value gR. Note that thisthreshold value can be a percentage of the maximum ofthe error indicators. Fig. 2 shows how one element (alsocalled parent) is refined into four smaller elements (alsocalled children).

Derefinement: We consider a parent element for dere-

finement if (A) all of its children are active, (B) the errorindicator of each of its children is less than a thresholdvalue gD, and (C) the element was not refined during thecurrent time step. For each parent element meeting theserequirements, an L2 projection is performed to retrieve

the degrees of freedom of the parent. Before actually doingthe derefinement, we check that the parent error indicatoris less than gR. If it is, we then derefine. If it is not, wedo not derefine.

4.3. Adaptive time stepping technique

For time strategy, we allow the time step to vary duringthe simulation. We uniformly divide the simulation interval(0, T) into whole steps of length Dti. At the start of eachwhole step, we try to compute the saturation for timeti + Dti, whereti is the current time. If the resulting satura-tion function is satisfactory, then we record it, calculate thenew pressure function, and proceed to the next whole step.

On the other hand, if the resulting saturation function isunsatisfactory, then we discard it and subdivide the wholestep into two half steps. We then compute the saturationfor the time at which the first half step ends. If the resultis acceptable, we proceed to the second half step, and if

its result is also acceptable, then we continue on to the nextwhole step. If one of the half steps does not yield satisfac-tory results, then we divide it into quarter steps, proceedingin the same manner as before, with the exception that weaccept the results of the quarter steps regardless of how

Fig. 13. Two-dimensional pressure contours at 7.5, 15, 22.5 and 30 days on an inhomogeneous medium.



12/16

satisfactory they are. For the numerical simulations in thispaper, a resulting saturation function was deemed unsatis-factory if the average saturation in any element exceededthe physically permissible range by more than 0.01; other-wise, it was considered satisfactory.

The main purpose of this time stepping technique is to

speed up computation without losing accuracy, thus toincrease the efficiency of the method.

5. Numerical examples

In the following simulations, we assume that the fluidand medium properties are

lo0:002 kg=ms; lw0:0005 kg=ms; /0:2;swt0 0:2; pwt0 3:45106 Pa;sin0:95; srw0:15; sro0;

pd5103

Pa:

The orders of approximation are discontinuous piecewiselinears for the saturation and discontinuous piecewise qua-dratics for the pressure. For the adaptive refinements andderefinements, we chose gR= 1 10

1 and gD= (1/3)gR.The well-known five-spot problem on homogeneous andheterogeneous media is first considered, then simulationswith highly varying permeability are presented.

5.1. Five-spot on homogeneous medium

The permeability tensor is K= 1011 dm2, whered is theKronecker delta tensor.Fig. 3shows the coarse mesh and

Fig. 14. Two-dimensional saturation contours at 7.5, 15, 22.5 and 30 days on an inhomogeneous medium.

Fig. 15. Permeability field and coarse mesh: permeability is 1011 in whiteregions and 1016 elsewhere.



13/16

the domainX embedded into the square (300,300)2. Fourproduction wells are located at each corner of the domain;the well bore corresponds to part ofCp1, where we assumethatpdir= 2.41 10

6 Pa. An injection well is located in theinterior of the domain; the well bore corresponds to theboundary Cs1 and the remainder ofCp1 and the pressure

is set pdir= 3.45 106

Pa. The flow of the phases is thusdriven by the gradient of pressure from the injection wellto the production wells.

The simulation is run for 52.5 days with a time stepvarying between 0.001875 days and 0.0075 days. Theparameters in(12)are r = 0 and = 1. Three dimensionalviews of contours of wetting phase pressure and saturationat selected times are shown inFigs. 4 and 5. In order to bet-ter analyze this example and because of the symmetry ofthe problem, we re-run the simulations on one quarter ofthe domain; this yields the quarter-five spot problem shownin Fig. 6. The injection well is at the left bottom cornerwhereas the production well is at the right top corner.

The domain is now embedded into (0,300)2.The contours of wetting phase pressure and saturation

at selected times are shown in Figs. 7 and 8. The locallyrefined and derefined meshes are also given on these fig-ures. One can conclude that the proposed error indicators

capture well the location of the front. As expected, themesh is more refined in the neighborhood of the saturationfront. It also appears that the mesh stays refined at theneighborhood of the injection well bore.

We compare the adaptive results with those obtained onthe coarse mesh refined uniformly three times. The pressure

and saturation contours are given inFigs. 9 and 10. Here,the time step varies between 0.015 days and 0.00375 days.The contours are similar to the adaptive ones. For bettercomparison, we show the pressure and saturation profilesalong the diagonal {(x,y):x= y} (seeFig. 11). Using adap-tive refinement and derefinement decreases significantly thecost of the computation, as shown in Table 1. The columnsfor AMR correspond to adaptively refined meshes, and thecolumns for UNI correspond to uniformly refined meshes.

We now compare the saturation and pressure profilesobtained by varying the parametersr and . In particular,we consider the cases (,r)2{(1,1),(1,105)}, which yieldthe NIPG method with small and large penalty values;

the cases (,r)2{(1,1),(1,105)}, which yield the SIPGmethod with small and large penalty values; the cases(,r)2{(0,1),(0,105)}, which yield the IIPG method withsmall and large penalty values and the OBB method usedabove. If the penalty value is small enough, i.e., r= 105,

Fig. 16. Two-dimensional saturation contours at 17.5, 35, 52.5 and 70 days: ( = 1, r = 0).



14/16

all methods produce identical solutions for both pressureand saturation. Fig. 12 shows the profiles obtained at 15and 30 days: solid lines correspond to small penalty valuesor the OBB method whereas dashed lines correspond tolarge penalty values. As expected, if the penalty valueincreases, numerical diffusion added by the jump term

appears in the solutions. The saturation fronts are slightlysmeared. It is also interesting to note that for a fixed pen-alty value, all three penalty methods produce the samesolutions.

Finally, we give inTable 2the total number of degrees offreedom for pressure and saturation equations and we showthat the numerical cost is comparable for all methods.

5.2. Five spot on heterogeneous medium

This simulation is identical to the one above except forthe permeability tensor. Here, K is discontinuous and isequal to 1015 dm2 in a small subdomain. In the rest of

domain, K= 1011 dm2. We present the contours of thepressure and saturation at different times in Figs. 13 and14in the case (,r) = (1,0). Clearly, the region of low per-meability is not invaded by the injected wetting phase. Thisshows that the scheme has very little numerical diffusion. Itis also interesting to note that the proposed method allows

for an arbitrary number of hanging nodes, without anyspecial care.

5.3. Highly varying permeability field

We consider a square domain (0, 400)2 with varying

permeability as shown in Fig. 15. The permeability is1011 Im2 except in several small regions where it is 105

times smaller (see[16]). The simulation is run for 70 days.The time step varies between 2.1875 103 days and8.75 103 days. The vertical boundaries correspond toCp1 where the same pressure pdir as in the previous exam-ples is imposed. The left vertical boundary correspondsto Cs1. We first consider the OBB method (= 1,r= 0).Saturation contours on adaptively refined meshes areshown in Fig. 16. The degrees of freedom are 8232,13,281, 17,070 and 19,131 for the respective times 17.5,35, 52.5 and 70 days. The figures show clearly that thereis very little numerical diffusion. For comparison, we show

the contours obtained on a uniform mesh, which corre-sponds to 38,400 degrees of freedom (see Fig. 17). Thecoarse mesh has been refined twice, and this produces acomputational time of 24 h on a single processor. Foranother level of refinement, the simulation would run for1 week.

Fig. 17. Two-dimensional saturation contours at 17.5, 35, 52.5 and 70 days on uniform meshes: ( = 1, r = 0).



15/16

We repeat the experiment for the NIPG method withthree choices of penalty r2{106, 105,1}. The saturationcontours at 35 days are shown in Fig. 18. Because of thehighly varying permeability field, the method is more sensi-tive to the choice of the penalty. The value r = 106 yieldsa comparable solution to the OBB method; there is very lit-

tle numerical diffusion. However, for r = 105, the wettingphase floods the region of lower permeability and forr = 1, the method is too diffusive to capture the barrierzones. Similar conclusions can be made with the othertwo methods. We show the saturation contours forr= 106 for IIPG and SIPG inFig. 19.

Fig. 18. NIPG Two-dimensional saturation contours at 35 days:r = 106, r = 105 and r = 1.

Fig. 19. Two-dimensional saturation contours at 35 days for r = 10

6: IIPG (left) and SIPG (right).



16/16

6. Conclusions

This paper present adaptivity techniques in space andtime. We show that the adaptive simulations are more effi-cient than the simulations obtained on uniform meshes andwith constant time step. A slope limiting technique for

meshes with several hanging nodes per face is defined.Numerical experiments show robustness of the proposedDG schemes on heterogeneous media. Comparisonsbetween SIPG, NIPG, IIPG and OBB methods are per-formed: if the penalty value is small enough, the resultingnumerical solutions are very similar. Increasing the penaltyvalue introduces numerical diffusion in the approxima-tions, in particular if the permeability field highly variesin space. Several future extensions are currently underinvestigation: for instance, we plan to extend our computa-tional results to three-dimensional problems using unstruc-tured tetrahedral meshes. We are also investigating theeffects of p-adaptivity on the accuracy and the computa-

tional efficiency of the scheme.

Acknowledgement

The first author is partially supported by a CRDF grantfrom the University of Pittsburgh and by a Brackenridgefellowship. The second author is supported by a NationalScience Foundation grant DMS-0506039.

References

[1] D.N. Arnold, An interior penalty finite element method withdiscontinuous elements, SIAM J. Numer. Anal. 19 (1982) 742760.[2] P. Bastian, High order discontinuous Galerkin methods for flow and

transport in porous media, Challenges in Scientific Computing CISC2002, vol. 35, 2003.

[3] P. Bastian, Discontinuous Galerkin methods for two-phase flow inporous media, Also technical report, 2004-28, IWR (SFB 359),Heidelberg University, submitted for publication.

[4] B. Riviere, Numerical study of a discontinuous Galerkin method forincompressible two-phase flow, in: Proceedings of ECCOMAS, 2004.

[5] R.H. Brooks, A.T. Corey, Hydraulic properties of porous media,Hydrol. Pap. 3 (1964).

[6] G. Chavent, J. Jaffre, Mathematical Models and Finite Elements forReservoir Simulation, North-Holland, 1986.

[7] B. Cockburn, C.-W. Shu, TVB RungeKutta local projection

discontinuous Galerkin finite element method for conservative lawsII: general framework, Math. Comput. 52 (1989) 411435.[8] B. Cockburn, C.-W. Shu, The local discontinuous Galerkin method

for time-dependent convectiondiffusion systems, SIAM J. Numer.Anal. 35 (1998) 24402463.

[9] C. Dawson, S. Sun, M.F. Wheeler, Compatible algorithms forcoupled flow and transport, Comput. Meth. Appl. Mech. Engrg. 193(2004) 25652580.

[10] L.J. Durlofsky, B. Engquist, S. Osher, Triangle based adaptivestencils for the solution of hyperbolic conservation laws, J. Comput.Phys. 98 (1992) 6473.

[11] Y. Epshteyn, B. Riviere, Fully implicit discontinuous Galerkinschemes for multiphase flow, Appl. Numer. Math., in press.

[12] A. Ern, J. Proft, A posteriori discontinuous Galerkin error estimatesfor transient convectiondiffusion equations, Appl. Math. Lett. 18 (7)(2005) 833841.

[13] O. Eslinger, Discontinuous Galerkin finite element methods appliedto two-phase, air-water flow problems, Ph.D. thesis, The Universityof Texas at Austin, 2005.

[14] R. Helmig, Multiphase Flow and Transport Processes in theSubsurface, Springer, 1997.

[15] H. Hoteit, Ph. Ackerer, R. Mose, J. Erhel, B. Philippe, New two-dimensional slope limiters for discontinuous Galerkin methods onarbitrary meshes, Int. J. Numer. Meth. Engrg. 61 (2004) 25662593.

[16] L.J. Durlofsky, Accuracy of mixed and control volume finite elementapproximations to Darcy velocity and related quantities, WaterResour. Res. 30 (4) (1994) 965973.

[17] D. Nayagum, G. Schafer, R. Mose, Modelling two-phase incom-pressible flow in porous media using mixed hybrid and discontinuousfinite elements, Comput. Geosci. 8 (1) (2004) 4973.

[18] J.T. Oden, I. Babuska, C.E. Baumann, A discontinuous hp finiteelement method for diffusion problems, J. Comput. Phys. 146 (1998)491519.

[19] B. Riviere, The dgimpes model in ipars: Discontinuous Galerkin fortwo-phase flow integrated in a reservoir simulator framework,Technical Report 02-29, Texas Institute for Computational andApplied Mathematics, 2002.

[20] B. Riviere, E. Jenkins, In pursuit of better models and simulations, oilindustry looks to the math sciences, SIAM News (January/February)(2002).

[21] B. Riviere, M.F. Wheeler, Discontinuous Galerkin methods for flowand transport problems in porous media, Commun. Numer. Meth.Engrg. 18 (2002) 6368.

[22] B. Riviere, M.F. Wheeler, Non conforming methods for transportwith nonlinear reaction, in: Proceedings of an AMS-IMS-SIAM JointSummer Research Conference on Fluid Flow and Transport inPorous Media: Mathematical and Numerical Treatment, 2002.

[23] B. Riviere, M.F. Wheeler, V. Girault, Improved energy estimates forinterior penalty, constrained and discontinuous Galerkin methods forelliptic problems. Part I, Comput. Geosci. 3 (1999) 337360.

[24] S. Sun, B. Riviere, M.F. Wheeler, A combined mixed finite elementand discontinuous Galerkin method for miscible displacement prob-lems porous media, in: Proceedings of International Symposium onComputational and Applied PDEs, 2002, pp. 321348.

[25] S. Sun, M.F. Wheeler, Symmetric and non-symmetric discontinuousGalerkin methods for reactive transport in porous media, SIAM J.Numer. Anal. 43 (1) (2005) 195219.

[26] S. Sun, M.F. Wheeler, Symmetric and nonsymmetric discontinuousGalerkin methods for reactive transport in porous media, SIAM J.Numer. Anal. 43 (1) (2005) 195219.

[27] M.F. Wheeler, An elliptic collocation-finite element method withinterior penalties, SIAM J. Numer. Anal. 15 (1) (1978) 152161.


Bonecos.pdf

Documents

Transcript of Bonecos.pdf