discontinuous galerkin methods for flow and transport problems in porous media

COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERINGCommun. Numer. Meth. Engng 2002; 18:63–68 (DOI: 10.1002/cnm.464)

Discontinuous Galerkin methods for .ow and transportproblems in porous media

B3eatrice Rivi5ere∗ and Mary F. Wheeler

The Center for Subsurface Modeling; Texas Institute for Computational and Applied Mathematics;The University of Texas at Austin; ACE 5.332; 201 East 24th Street; Austin TX 78712; U.S.A.

SUMMARY

This work presents a new scheme based on discontinuous approximation spaces for solving the miscibledisplacement problem in porous media. Numerical comparisons are made between this scheme and thewell-known mixed =nite element and higher-order Godunov methods. The simulations clearly show theadvantages of the discontinuous Galerkin methods for stable or unstable .ow. Copyright ? 2001 JohnWiley & Sons, Ltd.

KEY WORDS: miscible displacement; higher-order schemes; discontinuous polynomials; unstructuredmeshes; unstable .ow

1. INTRODUCTION

We consider the displacement of one incompressible .uid by another in a porous medium Bin R2. The invading and the displaced .uids are referred to as the solvent and the resident.uid, respectively. Let J denote the time interval (0; Tf]. The classical equations governingthe miscible displacement in B over J are

−∇ ·(K�(c)

∇p)≡∇ · u=0 in B× J

@c@t

+∇ · (uc −D(u)∇c) = R(c) in B× J

where the dependent variables are p, the pressure in the .uid mixture, and c, the fractionvolume of the solvent in the .uid mixture. The permeability K of the medium measures theresistance of the medium to .uid .ow, � is the viscosity of the .uid mixture, u represents

∗Correspondence to: B. Rivi5ere; TICAM, The University of Texas at Austin; ACE 5.332; 201 East 24th Street;Austin; TX 78712; U.S.A.

Received December 2000Copyright ? 2001 John Wiley & Sons, Ltd. Accepted September 2001

64 B. RIVI 5ERE AND M. F. WHEELER

the Darcy velocity, is the porosity of the medium and D(u) is the coeKcient of moleculardiLusion and mechanical dispersion that depends on u in a non-linear fashion. A survey onsimulating miscible displacement can be found in Reference [1].

The boundary of the domain is decomposed into a Dirichlet part ND and a Neumann part NN

such that ND∩NN = ∅ and ND ∪NN = @B. We also de=ne the in.ow part N− ≡ {x∈ @B : u ·]¡0}and the out.ow part N+ ≡ {x∈ @B : u ·]¿0}, where ] denotes the unit outward normal vectorto @B. We assume Dirichlet and Neumann boundary conditions for the pressure and Neu-mann and mixed boundary conditions for the concentration. The viscosity of the .uid mixtureis assumed to follow the quarter-power mixing law, commonly applicable to hydrocarbonmixtures [2]: �(c)= (c�−0:25

s +(1−c)�−0:250 )−4 where �s (resp. �0) is the viscosity of the sol-

vent (resp. resident .uid). The stability of the .ow is characterized by the mobility ratio, i.e.the ratio of the viscosity of the resident .uid to the viscosity of the solvent. Instabilities in the.ow will grow if the mobility ratio is larger than unity. In that case, protusions referred to asviscous =ngering develop through the resident .uid. Another important physical parameter isthe Peclet number that quanti=es the convective eLects with respect to the dispersive eLects.

2. NOTATION AND SCHEME

Let EFh ={EF1 ; E

F2 ; : : : ; E

FNFh

}and ETh = {ET1 ; ET2 ; : : : ; ETNTh } be two non-degenerate subdivisions

of B, that consist of triangles or quadrilaterals. The edges of EFh (resp. ETh ) are denoted byeFk (resp. eTk ). Let PFh (resp. PTh ) denote the number of interior edges in EFh (resp. ETh ). Oneach edge eFk (resp. eTk ), a unit normal vector ]Fk (resp. ]Tk ) is arbitrarily =xed, except on theboundary @B where it coincides with the outward unit normal vector. The direction of ]Fkuniquely de=nes the jump of the function : []=|E1 − |E2 if ]Fk is from E1 to E2. Theaverage of is de=ned by {}=(|E1 +|E2)=2. For rF and rT positive integers, one de=nesthe following discrete spaces:

DrF (EFh )= {v : v|EFj ∈PrF (EFj ); ∀j}; DrT (ETh )= {v : v|ETj ∈PrT (ETj ); ∀j}

We now formally de=ne two bilinear forms

aNS(c;p; v) =NFh∑j=1

∫EFj

K�(c)

∇p∇v−PFh∑k=1

∫eFk

{K�(c)

∇p · ]Fk}

[v]+PFh∑k=1

∫eFk

{K�(c)

∇v · ]Fk}

[p]

− ∑eFk ∈ ND

∫eFk

(K�(c)

∇p · ]Fk)v+

∑eFk ∈ND

∫eFk

(K�(c)

∇v · ]Fk)p

bNS(u; c; w) =NTh∑j=1

∫ETj

D(u)∇c∇w −∫ETj

cu · ∇w +PTh∑k=1

∫eTk

c∗u · ]Tk [w]

+∑eTk ∈N+

∫eTk

cu · ]Tk w −PTh∑k=1

∫eTk

{D(u)∇c · ]Tk }[w] +PTh∑k=1

∫eTk

{D(u)∇w · ]Tk }[c]

Copyright ? 2001 John Wiley & Sons, Ltd. Commun. Numer. Meth. Engng 2002; 18:63–68

DG METHODS FOR MISCIBLE DISPLACEMENT 65

where c∗ is the upwind value of the concentration on a given edge. We also de=ne thefunctionals

LF(v)=∑eFk ∈ND

∫eFk

(K�(c)

∇v · ]Tk)p0; LT (w)=

∫BR(c)w − ∑

eTk ∈N−

∫eTk

cinu · ]Tk w

where p0 is the Dirichlet datum for pressure and cin is the in.ow datum for concentra-tion. The continuous in time discontinuous Galerkin (DG) [3] method is given by the map(PDG; CDG) : [0; T ]→DrF (EFh )×DrT (ETh ) determined by the relations for any t in J

aNS(CDG(t);PDG(t); v) = LF(v); ∀v∈DrF (EFh ) (1)∫B@CDG

@tw + bNS(UDG(t);CDG; w) = LT (w); ∀w∈DrT (ETh ) (2)

The initial concentration CDG(0) is the L2 projection of the initial concentration and the Darcyvelocity is de=ned by UDG = − (K=�(CDG))∇PDG: We propose a time-stepping procedure thatre.ects the fact that the velocity =eld varies more slowly in time than the concentration forreasonable physical data. Thus, the pressure time step RtF¿0 will be chosen to be m timeslarger than the concentration time step RtT¿0. The procedure has been shown to be eKcientfor other =nite element methods applied to the miscible displacement problem [4]. We denotet iT = iRtT and t jF = jRtF . We now describe the algorithm for advancing of one pressure timestep. We assume that t iT = t

jF for some =xed indices i and j and that the approximation CDG(t iT )

is known. Then, the pressure at time t jF can be calculated as the solution of (1) for t= t iT . Thediscretization in time of the concentration equation for t i+#T , where 16#6m, is accomplishedby deriving a modi=ed version of the standard backward-diLerence scheme procedure

∫BCDG(t i+#T )− CDG(t i+#−1

T )RtT

w + bNS(UDG# ;CDG(t i+#T ); w)=LT (w); ∀w∈DrT (ETh )

where UDG# is a linear extrapolation of UDG(t jF) and UDG(t j−1

F ). For the =rst pressure step,this extrapolation is not valid and we use a predictor–corrector technique. The use of slopelimiters [5] is also needed to prevent from numerical overshoots and undershoots to occur inthe neighbourhood of the concentration front.

3. NUMERICAL EXPERIMENTS

In all experiments, the permeability =eld is randomly generated on the coarse mesh. We solvefor quadratic approximations of the pressure and the concentration.

We =rst consider the case where both .ow and transport equations are solved on structuredmeshes. In that case, we compare our simulations with those obtained from the ParallelSubsurface Simulator (Parssim) developed at the University of Texas at Austin [6]. In Parssim,.ow is simulated using the mixed =nite element method and transport is simulated using ahigher-order Godunov method. Figure 1 shows the permeability =eld and the concentrationfront obtained with the DG method in the case of mobility ratio 1 and a low Peclet number.



Figure 1. (a) Permeability =eld on coarse mesh; and (b) DG Concentration front on mesh re=ned twicefor mobility ratio 1 and Peclet number 40.

PARSSIM h3-h3

DG h0-h2

DG h1-h1

DG h1-h2

PARSSIM h3-h3

DG h0-h2

DG h1-h1

DG h1-h2

(a) (b)

Figure 2. Comparisons between DG and Parssim concentration isocontours for mobility ratio 1 andPeclet number: (a) 40; and (b) 1600.

The isocontours c=0:25 obtained with DG and Parssim for low and high Peclet number areshown in Figure 2. The legend ‘DG h0-h1’ means that .ow is solved on the coarse mesh andtransport is solved on the mesh re=ned once. The DG concentration fronts are comparableto those obtained by Parssim even though coarser meshes have been used for DG. We alsonote that DG velocities are accurate enough to be computed on a coarser mesh than DGconcentrations. As we increase the mobility ratio (see Figure 3), instabilities in the .ow yieldmore =ngers in the concentration front. We also observe that DG velocities computed onthe coarse mesh produce a much more detailed concentration front than Parssim velocitiescomputed on the coarse mesh.

We then consider the case where .ow is solved on an unstructured mesh and transport issolved on a structured mesh. The motivation for solving each equation on a diLerent meshlies in the fact that in realistic media, the permeability =eld varies greatly in space in avery unstructured manner. However, the porosity of the medium is assumed to be uniform,and structured meshes are preferred for solving the concentration equation. Comparisons withParssim are not possible since Parssim does not handle unstructured meshes. Figure 4 showsthe permeability =eld and the pressure =eld obtained in the case of mobility ratio 1. We


DG METHODS FOR MISCIBLE DISPLACEMENT 67

DG h0-h0

PARSSIM h0-h0

DG h0-h1

PARSSIM h1-h1

PARSSIM h0-h0

DG h0-h0

DG h0-h1

PARSSIM h1-h1

(a) (b)

Figure 3. Comparisons between DG and Parssim concentration isocontours for mobility ratio 10 andPeclet number: (a) 40; and (b) 1600.

Figure 4. (a) Permeability =eld on unstructured coarse mesh; and (b) DGPressure =eld on coarse unstructured mesh.

DG h2-h2

DG h1-h1

DG h1-h2

DG h2-h2

DG h1-h1

DG h1-h2

(a) (b)

Figure 5. Isocontours of the DG concentration for mobility ratio 1 and forPeclet number: (a) 40; and (b) 1600.

compare our numerical results to the DG solution computed on the mesh re=ned twice forboth pressure and concentration, which we refer to as the +ne solution. As the meshes forconcentration and pressure are successively re=ned, we observe that the isocontours convergeto the +ne solution (see Figure 5(a)). In the case of high Peclet number, the front has moreprotusions (see Figure 5(b)). We repeat the experiments for unstable .ow (see Figure 6) andthe same phenomena are observed. However, the viscous =ngers are more pronounced.



DG h2-h2

DG h1-h1

DG h1-h2

DG h2-h2

DG h1-h1

DG h1-h2

(a) (b)

Figure 6. Isocontours of the DG concentration for mobility ratio 10 and forPeclet number: (a) 40; and (b) 1600.

4. CONCLUSIONS

In this work, we have shown numerically that the DG method is well suited for .uid .owproblems in porous media and in particular that it is competitive with other locally con-servative methods for solving the miscible displacement problem. First, DG velocities areaccurate enough so that coarser meshes can be used for solving the .ow equation. Second,the DG method can capture the instabilities of the .ow. Finally, the DG method can handleunstructured meshes in an easy and natural way.

REFERENCES

1. Douglas J. The numerical simulation of miscible displacement in porous media. Computational Methods inNonlinear Mechanics, Oden JT (ed.). North-Holland; Amsterdam, 1980; 225–237.

2. Koval EJ. A method for predicting the performance of unstable miscible displacement in heterogeneous media.Society of Petroleum Engineers Journal 1963; 3:145–154.

3. Rivi5ere B, Wheeler MF, Girault V. A priori error estimates for =nite element methods based on discontinuousapproximation spaces for elliptic problems. SIAM Journal on Numerical Analysis 2001; 39(3):902–931.

4. Douglas J, Ewing RE, Wheeler MF. A time-discretization procedure for a mixed =nite element approximation ofmiscible displacement in porous media. RAIRO Numerical Analysis 1983; 17(3):249–265.

5. Cockburn B, Shu CW. The Runge–Kutta discontinuous Galerkin method for conservation laws V. Journal ofComputational Physics 1998; 141:199–224.

6. Arbogast T. User’s Guide to Parssim1: the Parallel Subsurface Simulator, Single Phase, Vol. 13. Texas Institutefor Computational and Applied Mathematics, Austin, 1998.


discontinuous galerkin methods for flow and transport problems in porous media

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