Comparison of DDFV and DG Methods for Flow in Anisotropic

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507 > EditorialJ. E. Roberts

515 > Modeling Fractures in a Poro-Elastic MediumUn modèle de fracture dans un milieu poro-élastiqueB. Ganis, V. Girault, M. Mear, G. Singh and M. Wheeler

529 > Modeling Fluid Flow in Faulted BasinsModélisation des transferts fluides dans les bassins faillésI. Faille, M. Thibaut, M.-C. Cacas, P. Havé, F. Willien, S. Wolf, L. Agelasand S. Pegaz-Fiornet

555 > An Efficient XFEM Approximation of Darcy Flows in Arbitrarily FracturedPorous MediaUne approximation efficace par XFEM pour écoulements de Darcy dansles milieux poreux arbitrairement fracturésA. Fumagalli and A. Scotti

565 > Hex-Dominant Mesh Improving Quality to Tracking Hydrocarbons in DynamicBasinsAmélioration de la qualité d’un maillage hexa-dominant pour la simulation del’écoulement des hydrocarburesB. Yahiaoui, H. Borouchaki and A. Benali

573 > Advanced Workflows for Fluid Transfer in Faulted BasinsMéthodologie appliquée aux circulations des fluides dans les bassins faillésM. Thibaut, A. Jardin, I. Faille, F. Willien and X. Guichet

585 > Efficient Scheme for Chemical Flooding SimulationUn schéma numérique performant pour la simulation des écoulements d’agentschimiques dans les réservoirs pétroliersB. Braconnier, E. Flauraud and Q. L. Nguyen

603 > Sensitivity Analysis and Optimization of Surfactant-Polymer Flooding underUncertaintiesAnalyse de sensibilité et optimisation sous incertitudes de procédés EOR de typesurfactant-polymèreF. Douarche, S. Da Veiga, M. Feraille, G. Enchéry, S. Touzani and R. Barsalou

619 > Screening Method Using the Derivative-based Global Sensitivity Indices withApplication to Reservoir SimulatorMéthode de criblage basée sur les indices de sensibilité DGSM : application ausimulateur de réservoirS. Touzani and D. Busby

633 > An Effective Criterion to Prevent Injection Test Numerical Simulation fromSpurious OscillationsUn critère efficace pour prévenir les oscillations parasites dans la simulationnumérique du test d’injectionF. Verga, D. Viberti, E. Salina Borello and C. Serazio

653 > Well Test Analysis of Naturally Fractured Vuggy Reservoirs with an AnalyticalTriple Porosity _ Double Permeability Model and a Global OptimizationMethodAnalyse des puits d’essai de réservoirs vacuolaires naturellement fracturésavec un modèle de triple porosité _ double perméabilité et une méthoded’optimisation globaleS. Gómez, G. Ramos, A. Mesejo, R. Camacho, M. Vásquez and N. del Castillo

673 > Comparison of DDFV and DG Methods for Flow in AnisotropicHeterogeneous Porous MediaComparaison des méthodes DDFV et DG pour des écoulements en milieuporeux hétérogène anisotropeV. Baron, Y. Coudière and P. Sochala

687 > Adaptive Mesh Refinement for a Finite Volume Method for Flow andTransport of Radionuclides in Heterogeneous Porous MediaAdaptation de maillage pour un schéma volumes finis pour la simulationd’écoulement et de transport de radionucléides en milieux poreux hétérogènesB. Amaziane, M. Bourgeois and M. El Fatini

701 > A Review of Recent Advances in Discretization Methods, a Posteriori ErrorAnalysis, and Adaptive Algorithms for Numerical Modeling in GeosciencesUne revue des avancées récentes autour des méthodes de discrétisation, del’analyse a posteriori, et des algorithmes adaptatifs pour la modélisationnumérique en géosciencesD. A. Di Pietro and M. Vohralík

731 > Two-Level Domain Decomposition Methods for Highly HeterogeneousDarcy Equations. Connections with Multiscale MethodsMéthodes de décomposition de domaine à deux niveaux pour les équationsde Darcy à coefficients très hétérogènes. Liens avec les méthodes multi-échellesV. Dolean, P. Jolivet, F. Nataf, N. Spillane and H. Xiang

753 > Survey on Efficient Linear Solvers for Porous Media Flow Models onRecent Hardware ArchitecturesRevue des algorithmes de solveurs linéaires utilisés en simulation deréservoir, efficaces sur les architectures matérielles modernesA. Anciaux-Sedrakian, P. Gottschling,J.-M. Gratien and T. Guignon

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D o s s i e rGeosciences Numerical Methods

Modélisation numérique en géosciences

Comparison of DDFV and DG Methods for Flowin Anisotropic Heterogeneous Porous Media

V. Baron1,2*, Y. Coudière3 and P. Sochala2

1 Laboratoire de mathématiques Jean Leray, 2 rue de la Houssinière, 44322 Nantes - France2 BRGM, 3 avenue Claude Guillemin, 45060 Orléans - France

3 IMB, 351 cours de la libération, 33405 Talence Cedex - Francee-mail: [email protected] - [email protected] - [email protected]

* Corresponding author

Resume — Comparaison des methodes DDFV et DG pour des ecoulements en milieu poreux

heterogene anisotrope — Nous presentons un travail preliminaire visant a simuler l’injection de

gaz dans les aquiferes profonds. Des ecoulements instationnaires monophasiques sont

consideres. Nous comparons des schemas volumes finis en dualite discrete (DDFV, Discrete

Duality Finite Volume) et de Galerkin discontinu (DG, Discontinuous Galerkin) utilises pour

discretiser le terme diffusif. Une formule de differentiation retrograde d’ordre deux est utilisee

pour la discretisation en temps. D’une part, les schemas DDFV sont simples a implementer,

preservent les proprietes physiques et presentent souvent une super convergence en norme L2.

D’autre part, les methodes DG sont flexibles, permettent des ordres de convergence

arbitrairement eleves et leur fondement theorique en font un choix adapte a de nombreux

problemes. La methode de Galerkin a penalisation interieure ponderee symetrique est choisie

dans ce travail. La precision et la robustesse de ces deux schemas sont testees et comparees sur

des cas tests varies, notamment en milieu heterogene anisotrope.

Abstract— Comparison of DDFV and DGMethods for Flow in Anisotropic Heterogeneous Porous

Media—We present a preliminary work to simulate gas injection in deep aquifers. Unsteady single-

phase flows are considered. We compare Discrete Duality Finite Volume (DDFV) and Discontinu-

ous Galerkin (DG) schemes applied to discretize the diffusive term. The second-order backward

differentiation formula is used for the time-stepping method. On the one hand, the DDFV methods

are easy to implement, ensure a preservation of physical properties and offer superconvergence in the

L2-norm on a regular basis. On the other hand, the DG methods are flexible, allow arbitrary order of

accuracy, and their ample theoretical foundation make them a reliable choice for many computa-

tional problems. We consider here the symmetric weighted interior penalty Galerkin method. Accu-

racy and robustness of these two schemes are tested and compared on various test cases, especially in

anisotropic heterogeneous media.

Oil & Gas Science and Technology – Rev. IFP Energies nouvelles, Vol. 69 (2014), No. 4, pp. 673-686Copyright � 2013, IFP Energies nouvellesDOI: 10.2516/ogst/2013157






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INTRODUCTION

Prediction of fluid flows in geological subsurface is a key

process in many applications like groundwater hydrol-

ogy, oil recovery, and more recently CO2 geological stor-

age and deep geothermal science. Several works are

devoted to design numerical schemes (listed below) for

simulating these types of flows. This article is a compar-

ison of two conservative methods which are accurate

(i.e: having a second-order convergence), robust in

anisotropic heterogeneous media (i.e: with a diffusivity

tensor inducing a space-dependent preferential direction

for the flow) and valid for general meshes (i.e: not only

for K-orthogonal meshes). We present a preliminary

work dedicated to single-phase flows. An extension of

this study is to consider two phases (gas and water) in

the frame of CO2 injection in deep aquifers.

A various choice of methods is available to approxi-

mate nonlinear diffusion equations. For instance to dis-

cretize the Richards equation, Celia et al. (1990) consider

Finite Element (FE), Narasimhan and Witherspoon

(1976) apply Finite Volume (FV) with two-point flux

approximation, Manzini and Ferraris (2004) use dia-

mond cell FV, and Knabner and Schneid (2002) employ

Mixed Finite Element (MFE). Several other FVmethods

and variants exist, including the MPFA, SUSHI and

VAG schemes, see Eymard et al. (2012) and references

therein for details. In this work, we propose an alterna-

tive approach by applying and comparing DDFV and

DG methods. The DDFV methodology was introduced

about ten years ago by Hermeline (2000). It has been

applied to a variety of problems over the past few years,

from isotropic scalar diffusion with Domelevo and

Omnes (2005) to electrocardiology, Coudiere et al.

(2009), among others. This FV method is a natural

choice because it has proven robustness when applied

to anisotropic and heterogeneous diffusion problems

on general, most notably non-matching and distorted

meshes. It is locally conservative and preserves the sym-

metry of the continuous problem, which allows efficient

iterative solvers to be employed. Besides it offers a

L2-norm superconvergence on various examples,

together with a very accurate appoximation of the gradi-

ent, as was noted in Herbin and Hubert (2008). In this

paper, we adapt a version of the DDFV scheme

described in Coudiere et al. (2009), Krell (2010) which

allows discontinuities of the permeability tensor along

faces. DG methods have appeared thirty years ago with

Lesaint and Raviart (1974) and are efficient to approxi-

mate equations present in engineering sciences. Advan-

tages of these methods are multiple especially the local

conservation (like FV and MFE). The order of accuracy

(like FE and MFE) can be different for each element of

the mesh, facilitating the p-refinement. The flexibility in

the use of non-matching meshes (like FV) is particularly

adaptated for the h-refinement. Several DGmethods can

be used for one-phase and two-phase flows in porous

media, such as the local DGmethod described in Bastian

et al. (2007), Fagherazzi et al. (2004) and the non-

symmetric or the symmetric interior penalty Galerkin

method used in Bastian (2003), Klieber and Riviere

(2006), Sochala et al. (2009). Here, we choose the Sym-

metric Weighted Interior Penalty DG method (SWIP),

because it preserves the natural symmetry in the discrete

diffusion operator and uses diffusion-dependent

weighted averages to reduce the amount of stabilization

required, Di Pietro and Ern (2012). Regarding time dis-

cretization, we propose to use the second-order back-

ward differentiation formula (BDF2) from Curtiss and

Hirschfelder (1952). This implicit scheme avoids the

CFL condition particularly restrictive for explicit

schemes when the diffusion term is present. Further-

more, BDF2 is more efficient than implicit Runge-Kutta

(DIRK) schemes, which require several stages (three for

the second-order version). Besides, the last stage of

DIRK schemes is ill-posed where the soil is saturated,

as was underlined in Sochala et al. (2009). Finally,

BDF2 is preferred to Crank-Nicolson which is not

L-stable, Hairer and Wanner (2010).

The aim of this work is to test and compare the

DDFV and SWIP schemes for solving flows in aniso-

tropic heterogeneous porous media. We propose to con-

sider the conservative form of the Richards equation in a

two-dimensional domain X:

@thðwÞ � r � ðKðwÞðrwþ ezÞÞ ¼ f in X � �0; T �w ¼ w0 in X� f0gw ¼ wD on @XD � �0; T ��KðwÞðrwþ ezÞ � nX ¼ vN on @XN � �0; T �

8>>><>>>:

ð1Þ

where T is the final time of simulation, @XD (resp. @XNÞthe boundary of X where a Dirichlet condition wD (resp.

Neumann condition vN ) is applied, and nX the outward

unit normal of @X ¼ @XD [ @XN. The vector ez ¼ ð0; 1Þtis the gravity contribution. The pressure head w is the

unknown and is related to the velocity v through Darcy’s

law vðwÞ ¼ �KðwÞðrwþ ezÞ. Two constitutive relation-

ships w 7! hðwÞ and w 7! KðwÞ are necessary to close

the model. Examples of water content h and conductivity

K are indicated in Sections 3.1 and 3.2. For the sake of

simplicity, we suppose that the source term f is equal to

zero (except for the validation test case described in

Sect. 3.1). We investigate three Test Cases (TC) to

674 Oil & Gas Science and Technology – Rev. IFP Energies nouvelles, Vol. 69 (2014), No. 4

compareDDFV and SWIPmethods in several configura-

tions. TC1 and TC2 are column infiltration problems in

an isotropic homogeneous soil. TC1 has an analytical

solution and permits to validate the convergence rate of

the methods. The stencil of the DDFV method produces

a sparsermatrix than SWIP. In return, the latter leads to a

narrower bandwith and smaller condition numbers. TC2

features a stiff pressure front which highlights significant

differences for coarse meshes. DDFV underestimates

the front propagation speed while SWIP presents non-

physical oscillations. TC3 is a quarter five-spot configura-

tion in an anisotropic heterogenous soil. The two

methods only differ in the evaluation of the mass error,

depending on the choice of the nonlinear solver tolerance.

The outline is as follows. Section 1 introduces the

space discretization associated with DDFV and SWIP

methods. Section 2 describes the time discretization per-

formed by the BDF2 formula and the Picard fixed point

algorithm used to solve the nonlinear discrete system.

Section 3 details the results on the three TC described

above. Conclusions and perspectives are drawn after.

1 SPACE DISCRETIZATION

1.1 Notations

Let fT hgh>0 be a shape-regular family of unstructured

meshes of X consisting for simplicity of affine triangles.

For an element s 2 T h, let @s denote its boundary and

ns its outward unit normal. The set F h of mesh faces is

partitioned into F ih [ FD

h [ FNh , where F i

h is the set of

internal faces and FDh (resp. FN

h ) is the set of Dirichlet

(resp. Neumann) faces. For a face r 2 F ih, there exist

sþ and s� in T h such that r ¼ @sþ \ @s�. We denote

by Fs � F h the set of the faces r such that

@s ¼ [r2Fsr, and by Fs � F h the set of the faces r that

share s as a vertex. We define nr as the unit normal to

r pointing from s� to sþ (Fig. 1).

The set Sh of mesh vertices is partitioned into

Sih [ SD

h [ SNh where Si

h are the interior vertices, SNh gath-

ers all the endpoints of the edges in FNh and SD

h are the

remaining boundary vertices. A secondary mesh is asso-

ciated to this set Sh as follows: any vertex s 2 Sih [ SN

h is

associated to a unique polygonal cell S whose vertices are

the centers of the triangles s and the midpoints of the

faces r that share s as a vertex; and which contains s(Fig. 2). Consequently, each face r is associated to its

two neighbouring cells sþ and s� and its two vertices,

denoted by sþ and s�. The following orientation assump-

tion is made for these notations:

detðxsþ � xs� ; xsþ � xs�Þ > 0

although the complete orientation is arbitrary and irrel-

evant in the sequel. Here, xs is the center of gravity of

s 2 T h and xs are the coordinates of s 2 Sh. We also

denote by xr the midpoints of the faces r 2 F h (Fig. 3).

Note that a face r � @X has only one neighbouring tri-

angle which is denoted by s� in the sequel, so that the

normal nr is also the unit normal to @X outward of X.The DDFV method couples, through the flux compu-

tations, two finite volume schemes written on a primary

mesh T h (considering the cells s as control volumes) and

on a secondary mesh Sh (considering the cells S as con-

trol volumes). Hence, the DDFV discrete unknown is a

set of degrees of freedom Wh, that eventually defines a

function wh (Sect. 1.2). The DG method directly defines

a discrete unknown function wh (Sect. 1.3) and is based

on a discrete variational formulation.

nσ

n τ +n τ

σ

τ −

−

−

τ +

+∂τ ∂τ

s +

s −

Figure 1

Basic notations.

S 1

S 2s 2

s 1

Primary mesh

Secondary mesh

Figure 2

Cells (in gray) of the secondary mesh used in the DDFV

method.

V. Baron et al. / Comparison of DDFV and DG Methods for Flow in Anisotropic Heterogeneous Porous Media 675

1.2 Discrete Duality Finite Volume Method

The DDFV method approximates the average of w on

each control volume s 2 T h and on each control volume

S associated to a vertex s 2 Sih [ SN

h . Hence, we define

the discrete DDFV unknown as:

Qh ¼def Wh ¼ ðws;wsÞs2T h;s2Sih[SN

h

n o

Several functions can be associated to these degrees of

freedom, for instance the usual piecewise constant func-

tion on each s 2 T h (or its sibling piecewise constant on

each S for s 2 Sh), but also the function wh that is piece-

wise linear on the triangles with base r and vertex xs (forall s 2 T h and r 2 Fs) and such that whðxsÞ ¼ ws and

whðxsÞ ¼ ws.

If we integrate Equation (1) on a control volume

s 2 T h we get:

d

dt

ZshðwÞ �

Xr2F�

s

ZrKðwÞðrwþ ezÞ � ns �

Xr2FN

s

ZrvN ¼ 0

where FNs ¼ Fs \ FN

h and F�s ¼ FsnFN

s . Note that the flux

�KðwÞðrwþ ezÞ � ns is continuous through each face rof s. The DDFV scheme reads, for any control volume

s 2 T h:

d

dtjsjhðwsÞ �

Xr2F�

s

Kðwr;sÞ rr;sWh þ ez� � � Nr �

Xr2FN

s

V r ¼ 0

ð2Þwhere wr;s (resp. rr;sWh) approximates w (resp. rw) onthe side s of the face r, the normal Nr ¼ jrjnr is the nor-mal to r outward of s accounting for the length of r and

V r approximatesRr vN. Note that in the TC presented in

Section 3, the boundary data is piecewise constant and

we can take exactly V r ¼Rr vN. There are several possi-

ble choices for the value wr;s. We take here:

wr;s ¼1

3ws þ ws� þ wsþð Þ

because it seems to provide the best accuracy and robust-

ness. For each vertex s 2 SDh , we choose to take

ws ¼ wDðxsÞ from the Dirichlet boundary data. For any

control volume S associated to a vertex s 2 Sih [ SN

h ,

the discrete equation is:

d

dtjSjhðwsÞ �

Xr2F�

s

ðKðwr;s�Þ rr;s�Wh þ ez� � � Ne�

þ Kðwr;sþÞ rr;sþWh þ ez� � � NeþÞ

�Xr2FN

s

ðKðwr;sÞ rr;sWh þ ez� � � Ne þ V rsÞ ¼ 0

ð3Þ

where FNs ¼ Fs \ FN

h and F�s ¼ FsnFN

s . In these summa-

tions, if r 2 F�s then the vertex s is identified to s�, the

neighbouring triangles are s� and the interface between

the control volumes associated to s� and sþ has two

parts eþ and e�, of which the corresponding normals

are Ne� ¼ ne� je�j as depicted in Figure 3. If r 2 FNs , then

there is only one triangle s neighbouring r but the flux

still has two terms, one along the edge e defined as in

the previous case (with the normal Ne ¼ nejej) and one

along the line ðxs; xrÞ � r. The number V rs approxi-

mates the integral of the Neumann boundary data along

this line. Given a face r 2 F ih, we denote by:

�Fr;s� ¼ �Kðwr;s�Þ rr;s�Wh þ ez� � � Nr

the fluxes out of s� and sþ through the face r and by:

Fe ¼ Kðwr;s�Þ rr;s�Wh þ ez� � � Ne�

þ Kðwr;sþÞ rr;sþWh þ ez� � � Neþ

the total flux from S� to Sþ through the interface

S� \ Sþ. Given a face r 2 FNh \ Fs and an endpoint s

of r,

Fe ¼ Kðwr;sÞ rr;sWh þ ez� � � Ne

is the flux from s to the opposite endpoint of r. The fluxFe is continuous through the interface between S� and

Sþ by construction. We ensure the continuity of the

fluxes Fr;s� through r by introducing an auxiliary

unknown wr used to compute the gradients rr;s�Wh

and rr;sþWh, following the DDFV strategy from

Coudiere et al. (2009):

τ− xτ−xτ+

τ+

s +

s −

e − e

+

xσ

ne−ne+

Figure 3

Notations for DDFV method.


rr;s�Wh ¼ 1

2Dr;s�ððwr � ws�ÞNr þ ðwsþ � ws�ÞNe�Þ

rr;sþWh ¼ 1

2Dr;sþððwsþ � wrÞNr þ ðwsþ � ws�ÞNeþÞ

where 2Dr;s� ¼ � detðxs� � xr; xsþ � xs�Þ, meaning that

Dr;s� are the areas of the triangles with base r and verti-

ces xs� . In the boundary case r 2 FNh , there is only one

gradient, namely rr;s�Wh. The auxiliary unknown is

then the unique solution to the linear equation

Fr;s� þ Fr;sþ ¼ 0 in the general interior case and to the

linear equation Fr;s� ¼ V r in the boundary case. Then

the flux between the primary cells s� is

Fr ¼ Fr;s� ¼ �Fr;sþ and the scheme (2, 3) reads:

d

dtjsjhðwsÞ �

Xr2F�

s

Fr �Xr2FN

s

V r ¼ 0

d

dtjSjhðwsÞ �

Xr2Fs

Fe �Xr2FN

s

V rs ¼ 0

In view of the expression of the fluxes and the gradi-

ents, we adopt some additional notations in the general

interior case. On each side s� of r, we define the follow-ing Gram matrices:

a� b�b� c�

� �¼def 1

2Dr;s�Kðwr;s�Þ

Nr

Ne�

� �Nr Ne�ð Þ

and a�, b� are defined similarly by:

a�b�

� �¼def Kðwr;s�Þez �

Nr

Ne�

� �

Hence, the equation in wr, Fr;s� þ Fr;sþ ¼ 0 now

reads: a�ðwr�ws�Þ þ b�ðwsþ�ws�Þþ a� ¼ aþðwsþ�wrÞþ bþðwsþ � ws�Þ þ aþ and can be easily solved. After-

wards, we replace wr by its expression and find that

the calculation of the fluxes Fr and Fe can be carried

out as follows:

Fr

Fe

� �¼ ArðWhÞ

wsþ � ws�

wsþ � ws�

� �� brðWhÞ

with

ArðWhÞ ¼a�aþa�þaþ

aþb�þa�bþa�þaþ

aþb�þa�bþa�þaþ

cþ þ c� � ðbþ�b�Þ2a�þaþ

0@

1A

brðWhÞ ¼ �aþa�þa�aþ

a�þaþ

b� þ bþ � bþ�b�a�þaþ

ðaþ � a�Þ

!

In the case of a face r 2 FNh \ s, we assume that the

neighbouring triangle s is s� and the equation on wr

reads a�ðwr � ws�Þ þ b�ðwsþ � ws�Þ þ a� ¼ V r. It is

solved easily and results in the expression:

Fe ¼ c� � b2�a�

� �wsþ � ws�ð Þ þ b� � b�

a�a� þ b�

a�V r

Lastly, if r 2 FDh , the quantity Fe does not need to be

defined, as only Fr adds a contribution to the system. In

this case, the auxiliary wr is chosen as wr ¼ 12 ðwsþ þ ws�Þ.

By assembling these local contributions, the semi-

discrete DDFV scheme finally reads:

Mhd

dtHðWhÞ þ AhðWhÞWh ¼ bhðWhÞ ð4Þ

where HðWhÞ ¼ hðWsÞ; hðWsÞð Þs2T h;s2Sih[SN

h, Mh is a diag-

onal mass matrix and AhðWhÞ is a symmetric and positive

matrix.

1.3 Discontinuous Galerkin Method

The Discontinuous Galerkin methods are based on

variational formulations (as for the classical finite

element). The discontinuous finite element space Vh is

defined as:

Vh ¼deff/ 2 L2ðXÞ; 8s 2 T h; /js 2 PpðsÞg

where PpðsÞ is the set of polynomials of degree less than

or equal to p on an element s. We observe that the func-

tions in Vh are not necessarily continuous. This fact is

exploited by selecting basis functions which are locally

supported in a single mesh element. As a consequence,

when a triangular mesh and a P1-Lagrange approxima-

tion are chosen, the unknowns are localized at the verti-

ces of each triangle as illustrated in Figure 4. A linear

approximation yields the same order of convergence as

DDFV, though higher-order elements could also be con-

sidered. We define the weighted average operator {}x,rand the jump operator [ ]r as follows: for a function

which is possibly two-valued on r:

fngx;r ¼def x�n� þ xþnþ and ½n�r ¼def n� � nþ

where n� ¼ njs� . For boundary faces, fngx;r ¼ njrand ½n�r ¼ njr. For vector-valued functions, average

and jump operators are defined componentwise.


For the Richards equation, the SWIP method can be

written as:

d

dt

ZXh whð Þ/þ ah wh;/ð Þ ¼ bh wh;/ð Þ

where for ðw;/Þ 2 Vh � Vh;

ah w;/ð Þ ¼defZXK wð Þrw � r/þ

Xr2F i

h[FDh

gcKdr

Zrw½ �r /½ �r

�X

r2F ih[FD

h

Zr

w½ �r KðwjsÞr/� �

x;r

�

þ /½ �r KðwjsÞrw� �

x;r

� nr

bhðw;/Þ ¼defZXr � ðKðwÞezÞ/

�Xr2FN

h

ðvN þ KðwjsÞez � nXÞ/

þXr2FD

h

Zr

�KðwjsÞr/ � nX þ gcKdr

/

� �wD

A specific choice of the weights is made to yield robust

error estimates with respect to the diffusivity:

x� ¼ dþKnd�Kn þ dþKn

and xþ ¼ d�Knd�Kn þ dþKn

with d�Kn ¼ K�nr � nr for r 2 F ih. In the penalty term, g is

a positive parameter (to be taken larger than a minimal

threshold depending on the shape-regularity of T h), and

dr is the diameter of the face r, i.e. the largest diameter of

the triangle(s) of which r is a face. The coefficient cK is

defined as:

cK ¼def2d�Knd

þKn

d�KnþdþKnif r 2 F i

h

dKn if r 2 FDh

8<:

with dKn ¼ Knr � nr. The above choice of x� and xþ

leads to the harmonic mean of the diffusivity tensor:

fKðwjsÞrwgx;r � nr ¼2d�Knd

þKn

d�Kn þ dþKnðrwjs� þ rwjsþÞ

Using harmonic means is particularly adapted at an

interface between poorly and highly conductive media.

In this case, the harmonic mean value of the diffusivity

is close to the value in the poorly conductive medium,

Di Pietro and Ern (2012).

Again, by assembling the local contributions, the

semi-discrete DG scheme takes the form of (4), where

Mh is a block diagonal matrix, Ah is a block matrix

and Wh is the set of the values of wh at each degree of

freedom.

2 TIME DISCRETIZATION ANDNONLINEAR RESOLUTION

2.1 Backward Differentiation Formula

The next step consists in approximating the time deriva-

tive in Equation (4). We propose the second order Back-

ward Differentiation Formula (BDF2):

d

dtyn ’ 1

dt3

2yn � 2yn�1 þ 1

2yn�2

� �

where the superscript n denotes the time level, and dt isthe constant time step, chosen such that T=dt is an inte-

ger. Substituting the last expression in Equation (4)

finally yields the following fully discrete scheme:

8n 2;3

2dtMhHðWn

hÞ þ AhðWnhÞWn

h

¼ bhðWnhÞ þ

2

dtMhHðWn�1

h Þ � 1

2dtMhHðWn�2

h Þ

The vector W1h is computed by the Crank-Nicolson

scheme.

2.2 Linearization Algorithm

The above is obviously a nonlinear system, and may thus

be solved through an iterative procedure, which we

DDFV DG - 1

Figure 4

Unknowns localization for each method.


describe in this section. The iteration level will be ref-

fered to as the superscript m. As suggested in Manzini

and Ferraris (2004), we use the Picard fixed point

method to obtain:

3

2dtMhHðWn;mþ1

h Þ þ AhðWn;mh ÞWn;mþ1

h

¼ bhðWn;mh Þ � 1

dt�2MhHðWn�1

h Þ þ 1

2MhHðWn�2

h Þ� �

To complete the linearization, we now use the follow-

ing Taylor series expansion:

HðyÞ ’ HðxÞ þ @H@W

ðxÞ � ðy� xÞ

which we apply with x ¼ Wn;mh and y ¼ Wn;mþ1

h . Finally,

the system to solve is:

3

2Mh@wHðWn;m

h Þ þ dtAhðWn;mh Þ

� �dWn;m

h

¼ dtbhðWn;mh Þ � dtAhðWn;m

h ÞWn;mh

� Mh3

2HðWn;m

h Þ � 2HðWn�1h Þ þ 1

2HðWn�2

h Þ� �

where dWn;mh ¼ Wn;mþ1

h �Wn;mh is the unknown of the sys-

tem. In this form, the algorithm produces at each time

level n, a sequence ðWn;mh Þm0 defined by the recurrence

relation Wn;mþ1h ¼ Wn;m

h þ dWn;mh , together with a

second-order initialization :Wn;0h ¼ 3Wn�1

h � 3Wn�2h þWn�3

h

for n 3, (and W1;0h ¼ W0

h;W2;0h ¼ 2W1

h �W0h). We use

the LU decomposition for solving each linear system

and our stopping criterion is jjdWn;mh jj2jjWn�1

h jj�12 �,

where � is the user-defined tolerance.

2.3 Mass Conservation Properties

The study of the mass evolution and repartition (i.e. the

separation over time of the mass variation in the domain,

the mass inflow and the mass outflow) helps to under-

stand the behavior of the system. It also allows to quan-

tify the mass defect generated by the linearization

algorithm. The volume of water in the domain X at time

ndt is obtained by integrating the volumetric water con-

tent in X:

Vn ¼defZXhðwn

hÞ

By using the BDF2 formula, we have:

3

2Vn � 2Vn�1 þ 1

2Vn�2 ¼ Fn

D þ FnN

� �dt þ �n ð5Þ

where the quantities FnD and Fn

N are defined as:

FnD ¼def �

Z@XD

v�;nh and FnN ¼def �

Z@XN

vnN

and �n stands for the numerical error in the resolution of

the nonlinear system. The reconstructed normal

velocity v�;nh on a Dirichlet face r is estimated from the

pressure:

v�;nh jr ¼ �Kðwnr;s�Þ rr;s�W

nh þ ez

� � � nr for DDFV

vðwnhjrÞ � nr þ gcK

drðwn

hjr � wDÞ for SWIP

(

The variation of the volume of water over the time

step ½ðn� 1Þdt; ndt� is deduced from a reformulation of

Equation (5), yielding:

Vn � Vn�1 ¼ UnD þ Un

N

� �dt þ en ð6Þ

where the flux Un 2 fUnD;U

nNg and the error en over the

time step ½ðn� 1Þdt; ndt� are defined as:

Un ¼def 23Fn þ 1

3Un�1; and en ¼def 2

3�n þ 1

3en�1

The initialization of the fluxes recursive formulas

depends on the scheme used at the first time step. For

the Crank-Nicolson scheme, the global mass conserva-

tion is written as:

V 1 � V 0 ¼ U1D þ U1

N

� �dt þ e1 with U1 ¼ 1

2F0 þ F1� �

ð7Þ

3 RESULTS

We propose three TC with increasing difficulties corre-

sponding to a more general form of the conductivity

defined as:

Kðw; xÞ ¼ qgmkðwÞKðxÞ

where kðwÞ is the relative permeability,KðxÞ the intrinsictensor permeability of the soil; q, g and m are respectivelythe fluid density, the gravity constant and the dynamic

viscosity. We first consider two downward infiltrations

in an isotropic homogeneous medium (i.e. K ¼ I). The

first TC presents an analytical solution to verify the the-

oretical convergence rates and the second TC is the prop-

agation of a stiff pressure front. We also study the


quarter five-spot problem in an anisotropic heteroge-

neous medium (i.e. K 6¼ I and is space-dependent).

3.1 Isotropic Homogeneous Validation Test Case

We propose a TC with an analytical solution to com-

pare the two methods in terms of matrix properties

and convergence. The domain is X ¼ ½0; 4� � ½0; 20�(in cm) and the final time T ¼ 2 min. The analytical

solution:

wðz; tÞ ¼ 20:4 tanh½0:5ðzþ t

12� 15Þ� � 41:1

is used to determine adequate source term and Dirichlet

boundary condition enforced on @X. The water contentand the conductivity are defined by Haverkamp’s consti-

tutive relationships which are derived in Haverkamp

et al. (1977):

hðwÞ ¼ hs � hr

1þ j~awjbþ hr and KðwÞ ¼ Ks

1þ j~Awjc

with parameters:

hs ¼ 0:287; hr ¼ 0:075; ~a ¼ 0:0271 cm�1; b ¼ 3:96;

Ks ¼ 9:44� 10�3 cm:s�1; ~A ¼ 0:0524 cm�1; c ¼ 4:74:

For a triangular mesh, the number of unknowns

Nu is:

N u ¼ N t þ N n � NDn for DDFV

3N t for DG� P1

(

where N t is the number of triangles, Nn the number of

nodes and NDn the number of nodes where a Dirichlet

condition is enforced (and which is equal to the number

of nodes located on @X). Consequently, the number of

unknowns for DDFV is about half the number of

unknowns for SWIP when the mesh is sufficiently fine

(i.e. N n � NDn ):

N t þ Nn � NDn ’ N t þ Nn ’ 3=2N t

because N t ’ 2Nn from the Euler relations. We have

therefore constructed two families of unstructured

meshes fMig16i66 and fMig16i66 respectively for DDFV

and SWIP, and such that the number of unknowns for

Mi and Mi are close. For each mesh, Table 1 refers to

Nn;N t and Nu. The number of nonzero elements nnz,

the mean number of nonzero elements per row nnz(¼ nnz=N uÞ and the bandwidth bw of the corresponding

matrix (obtained after a reverse Cuthill–McKee order-

ing, Cuthill and McKee (1969)) are also indicated. The

first remark is that the number of nonzero elements is

different for each method since it depends on the stencil.

By considering fine meshes (typically M5, M6, M 5 and

M 6), we have for DDFV:

nnz ’ 7N t þ 13N n ) nnz ’ 7N t þ 13N n

N t þ Nn � NDn

’ 9

while we have for SWIP:

nnz ¼ 9ð5N t � 2N n þ 2Þ)nnz ¼ 3ð5N t � 2N n þ 2ÞN t

’ 12

The second remark is that the bandwidth is smaller for

SWIP which has a more simple connectivity graph due to

the block structure of its matrix. To sum up, DDFV

requires less storage capacity whereas SWIP seems more

efficient for solving each linear system.

Table 2 presents the errors on the hydraulic head ew;Xand on the velocity ev;X for all the meshes:

ew;X ¼max

16n6NT

jjwn � wnhjjL2ðXÞ

max16n6NT

jjwnjjL2ðXÞ; ev;X ¼ jjv� vhjjL2ðX�½0;T �Þ

jjvjjL2ðX�½0;T �Þ

TABLE 1

TC1 – meshes used for DDFV on the top and SWIP on the bottom

Mesh Nn N t Nu nnz nnz bw

M1 28 34 42 234 5.6 6

M2 79 118 159 1 127 7.1 16

M3 253 430 609 4 897 8.0 34

M4 917 1 688 2 461 21 009 8.5 58

M5 3 380 6 474 9 570 837 42 8.8 145

M6 13233 25 896 39 031 348595 8.9 267

M1 13 16 48 504 10.5 12

M2 40 54 162 1 728 10.7 15

M3 128 204 612 6 894 11.3 24

M4 464 826 2 478 28 836 11.6 57

M5 1 702 3 198 9 594 113292 11.8 96

M6 6 591 12 780 38 340 456480 11.9 196


For the two methods, the results confirm theoretical

estimations since a second-order convergence is

observed on the hydraulic head and a first-order conver-

gence is verified on the velocity.

Table 3 reports the mean number N it of iterations in

the linearization algorithm and the mean condition

number j12:

N it ¼ 1

NT

Xn

Nnit and j12 ¼

Xn

Nnit

!�1 Xn;m

kn;mmax

kn;mmin

!

where Nnit is the number of iterations performed at time

level n and kn;mmax (resp. kn;mmin) is the maximal (resp. mini-

mal) eigenvalue of the matrix at the m-th nonlinear

iteration of the n-th time step. We observe that the mean

condition number is inversely proportional to the mesh

size h. This is consistent with a finite difference result

which states that the condition number j2 is propor-

tional to h�2. Furthermore, the SWIP method signifi-

cantly reduces ew;X (by a factor of 1.5) and j12 (by a

factor of 6) obtained with the most classical symmetric

interior penalty Galerkin method (which uses

xþ ¼ x� ¼ 0:5). To balance the various error sources

(due to space and time discretizations as well as nonlin-

ear systems resolution), an adaptive inexact Newton

method and adaptive time-stepping based on a posteriori

estimation could be implemented, as was done in Ern

and Vohralık (2010).

3.2 Isotropic Homogeneous Stiff Case

This TC is a stiff infiltration problem proposed by Celia

et al. (1990). The domain is X ¼ ½0; 20� � ½0; 100� (in cm)

and the final time is T ¼ 48 h. A constant initial condi-

tion is considered. A Dirichlet condition is imposed on

the top T and the bottom B of the column. An homoge-

neous Neumann condition (corresponding to a zero flux)

is imposed on the lateral parts L of the domain (Fig. 5):

w0 ¼ �10m in X

wD ¼ �10m on B��0;T�wD ¼ �75 cm on T ��0;T�vðwÞ � nX ¼ 0 on L��0;T�

8>>><>>>:

TABLE 2

TC1 – convergence results for DDFV on the top and SWIP on the

bottom

Mesh dt ew;X ev;X

Error Rate Error Rate

M1 8 1.00e-2 1.15e-1

M2 4 3.34e-3 2.46 4.18e-2 2.26

M3 2 1.00e-3 1.90 1.91e-2 1.23

M4 1 2.51e-4 2.23 9.41e-3 1.14

M5 1/2 6.29e-5 2.03 4.59e-3 1.05

M6 1/4 1.58e-5 1.99 2.28e-3 1.01

M1 8 4.51e-2 3.25e-1

M2 4 4.40e-3 2.60 1.17e-1 0.70

M3 2 1.10e-3 2.00 5.91e-2 0.99

M4 1 2.70e-4 2.17 2.94e-2 1.08

M5 1/2 6.41e-5 2.26 1.42e-2 1.14

M6 1/4 1.59e-5 1.92 7.09e-3 0.96

TABLE 3

TC1 – mean number of iterations and mean condition number

Mesh DDFV SWIP

N it j12 N it j12

1 2.9 65.1 2.3 2.0

2 1.8 120.5 1.2 2.1

3 1.0 203.6 1.1 4.6

4 1.0 432.1 1.0 11.6

5 1.0 813.7 1.0 24.8

6 1.0 1 625.4 1.0 54.5

1 m

ψD

vN = 0

•(0,0)= −10 m

ψD = −75 cm

Figure 5

TC2 – Polmann test case.


The water content and the conductivity are defined by

Van Genuchten’s constitutive relationships (van

Genuchten, 1980), and plotted in Figure 6:

hðwÞ ¼ ðhs � hrÞð1þ ðnjwjÞbÞc

þ hr

KðwÞ ¼ Ksð1� ðnjwjÞb�1ð1þ ðnjwjÞbÞ�cÞ2

ð1þ ðnjwjÞbÞc2

with parameters:

hs ¼ 0:368; hr ¼ 0:102; Ks ¼ 9:22� 10�3 cm:s�1;

n ¼ 0:0335 cm�1; b ¼ 2; c ¼ 1� b�1:

The evoked ‘‘stiffness’’ of this TC is related to the

strong constrained overpressure (equal to 9:25m)

imposed on the top of the column. A sharp vari-

ation of the conductivity ðKð�75 cmÞ=Kð�10 mÞ ¼8:92� 104Þ is induced, Figure 6.

We focus here on the horizontal mean value of the

pressure head wh zð Þ:

8z 2 ½0; 100�; whðzÞ ¼1

20

Z 20

0whðx; zÞdx

Figure 7 plots whðzÞ at 24 h and 48 h obtained with the

meshes (M4 -M 4), (M5 -M 5) and (M6 -M 6). The DDFV

pressure profiles are depicted by solid lines and the SWIP

pressure profiles are depicted by dashed lines. Each

method presents one drawback when coarse meshes

are used. For DDFV, a delay of the pressure front is

observed, especially when M4 is employed since a signif-

icant shift is observed between the two methods. For

SWIP, non-physical oscillations appear and could be

reduced with the help of a slope limiter as in Cockburn

and Shu (1998). In this stiff case, only DDFV satisfies

the discrete maximum principle whereas SWIP has a bet-

ter estimation of the propagation speed of the pressure

front. When the meshes are sufficiently fine, the two

methods yield the same pressure in agreement with Celia

et al. (1990) and Manzini and Ferraris (2004).

3.3 Anisotropic Heterogeneous Five-Spot Problem

This TC is inspired by the quarter five-spot problem

studied in Simmons et al. (1959), which reproduces an

elementary cell of a periodic network consisting of

sources and sinks. The domain X ¼ ½0; 1�2 (in m) is

divided into four parts (Fig. 8):

X1 ¼ X \ xþ z 6 0:5f gX2 ¼ X \ f0:5 < xþ z 6 1gX3 ¼ X \ 1 < xþ z 6 1:5f gX4 ¼ X \ f1:5 < xþ zg

The soil properties are the same as those of the previ-

ous TC except for the piecewise constant intrinsic perme-

ability:

KðxÞ ¼X4i¼1

1XiðxÞRxiDRtxi

where 1XiðxÞ is the indicator function of the set Xi, D is a

diagonal matrix and Rxi is the rotation matrix associated

to Xi:

D ¼ 1 0

0 10�3

�and Rx ¼ cosðxÞ � sinðxÞ

sinðxÞ cosðxÞ

�

Pressure head

Wat

er c

onte

nt

−500 −400 −300 −200 −1000.10

0.15

0.20

0.25

Pressure head

Con

duct

ivity

−500 −400 −300 −200 −1000

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0x 10−5

Figure 6

TC2 – constitutive curves hðwÞ and KðwÞ.


The angles of rotation are:

x1 ¼ p=4; x2 ¼ 0; x3 ¼ p=2; x4 ¼ p=4

The matrix Rtx denotes the transpose of Rx. This per-

meability induces a preferential direction for the flow on

each part of the domain as illustrated in Figure 8.

The final time is T ¼ 6 h. An hydrostatic initial condi-

tion is considered, an homogeneous Neumann condition

is imposed on the boundary of the domain except on the

top right corner C where an incoming flux vN is enforced

ez

eΩ1 Ω2

Ω3

Ω4

Most permeable direction

•(0,0)

(1,1)

Source

Sink

Figure 8

TC3 – five-spot problem in anisotropic heterogeneous

media.

0

20

40

60

80

100

0 20 40 60 80 100

Figure 9

TC3 – example of anisotropic heterogeneous mesh.

M4 - M 4

M5 - M 5

M6 - M 6

0 20 40 60 80 100−1 200

−1 000

−800

−600

−400

−200

0H

ydra

ulic

hea

d

Z

0 20 40 60 80 100−1 200

−1 000

−800

−600

−400

−200

0

Hyd

raul

ic h

ead

Z

0 20 40 60 80 100−1 200

−1 000

−800

−600

−400

−200

0

Hyd

raul

ic h

ead

Z

Figure 7

TC2 – pressure head whðzÞ at 24 h and 48 h obtained with dif-

ferent meshes for DDFV (solid line) and SWIP (dashed line).


and on the line L ¼ ½0; 0:025� � f0g where a zero pres-

sure condition is applied:

w0 ¼ �z in X

wD ¼ 0 on L��0;T�vðwÞ � nX ¼ vN on C��0;T�vðwÞ � nX ¼ 0 on C��0;T�

8>>><>>>:

where C ¼ f1g � ½0:975; 1� [ ½0:975; 1� � f1g and

C ¼ @X\ðL [ CÞ. The function vN (in cm:s�1) is defined as:

vNðtÞ ¼�5� 10�3 t

1 800 if t 0:5 h

�5� 10�3 if 0:5 h < t 6 4 h

0 if t > 4 h

8><>:

The condition vN ¼ �5� 10�3 cm:s�1 on C corre-

sponds to a water injection of 9 kg per hour.

Figure 9 shows that the triangles never cross the inter-

face between each subdomain. The meshes are obtained

from the FreeFrem� software, which allows to define

anisotropic heterogeneous metrics. Figure 10 plots

isolines of the overpressure wTh � w0

h at the final time of

the simulation. The two methods yield the same results

for fine meshes (8 760 triangles for DG, 16 446 for

DDFV). Concerning the convergence of the solution as

h tends to 0, SWIP needs further refinement than DDFV

when isotropic meshes are used. Therefore, DDFV is less

sensitive than SWIP to the choice of the mesh when an

anisotropic permeability is considered.

Figure 11 and Figure 12 present results on mass con-

servation. Multiplying Equations (6) and (7) by the

water density q, and summing over the time intervals

in ½0; ndt� leads to:qðVn � V 0Þ

�Mn

¼Xni¼1

qdtUiN

Miin

þXni¼1

qdtUiD

Miout

þXni¼1

qei

Ei

�Mn is the total mass variation over the time interval

½0; ndt� and the quantitiesPn

i¼1Miin,Pn

i¼1Miout andPn

i¼1jEij are respectively the total water inflow, the total

8 > < > : nn f

0 1000

100

X-axis

Y-a

xis

10

20

30

40

50

0 1000

100

X-axis

Y-a

xis

10

20

30

40

50

b)

a)

Figure 10

TC3 – isolines of overpressure at 6 h for a) DDFV and

b) SWIP.

0 1 2 3 4 5 6−30

−20

−10

0

10

20

30

40

Time

Mas

s

ΣMinn

ΣMnout

ΔMn

Figure 11

TC3 – mass repartition for DDFV (solid lines) and SWIP

(dashed lines).


water outflow and the total mass balance defect cumu-

lated at time ndt. The quantities �Mn;Pn

i¼1Miin andPn

i¼1Mnout are close for the two methods and three phases

are clearly recognizable in Figure 11. The first phase

½0; 1 h� corresponds to the soil saturation: the total mass

variation equals the total water inflow. The second phase

½1 h; 4 h� features soil saturation and drainage : the total

mass variation increases more slowly than during the

previous phase because exfiltration occurs. The last

phase ½4 h; 6 h� is the soil drainage only since the injectionis stopped, inducing a diminution of the total mass var-

iation.

The total mass balance defect (or mass error)Pn

i¼1jEijis plotted in Figure 12, where various tolerances � in the

linearization algorithm are tested. When � ¼ 10�3, the

defects look similar, but lowering the tolerance has no

improving effect with SWIP. Mass error can be reduced

by using higher order approximations instead. Mean-

while, about two orders of magnitude can still be gained

with DDFV before convergence, which is achieved for

� ¼ 10�5.

CONCLUSION

We presented a comparison between a discontinuous

finite element method and a more recent finite volume

scheme on various test cases. As was presented in

Section 1, the approaches are quite different and the

numerical results highlight different behaviors. First,

DDFV provides a sparser structure than SWIP which

produces in return a narrower bandwith and a better

condition number. In the stiff case, DDFV verifies the

discrete maximum principle while SWIP evaluates better

the speed of propagation for coarse meshes. Finally, in

the quarter five-spot problem, the two methods are in

accordance for the overpressure and the mass repartition

over time. The mass balance defect can be considerably

lowered with DDFV by requiring a more restrictive tol-

erance in the Picard algorithm.

REFERENCES

Bastian P. (2003) Higher order discontinuous Galerkin meth-ods for flow and transport in porous media, Bansch E. (ed.),Challenges in Scientific Computing - CISC 2002, pp. 1-22.

Bastian P., Ippisch O., Rezanezhad F., Vogel H.J., Roth K.(2007) Numerical simulation and experimental studies ofunsaturated water flow in heterogeneous systems, RannacharR., Warndz J. (eds), Reactive Flows, Diffusion and Transport,pp. 579-597.

Celia M.A., Bouloutas E.T., Zarba R.L. (1990) A generalmass-conservative numerical solution for the unsaturatedflow equation, Water Resources Research 26, (7), 1483-1496.

Cockburn B., Shu C.W. (1998) The local discontinuousGalerkin method for time-dependent convection-diffusion sys-tems, SIAM Journal on Numerical Analysis 35, (6), 2440-2463.

Coudiere Y., Pierre C., Rousseau O., Turpault R. (2009) A 2D/3D discrete duality finite volume scheme. Application to ECGsimulation, IJFV 6, (1), 1-24.

Curtiss C.F., Hirschfelder J.O. (1952) Integration of stiff equa-tions, Proceedings of the National Academy of Sciences of theUnited States of America 38, (3), 235.

Cuthill E., McKee J. (1969) Reducing the bandwidth of sparsesymmetric matrices, Proceedings of the 1969 24th national con-ference, ACM, pp. 157-172.

Di Pietro D.A., Ern A. (2012)Mathematical Aspects of Discon-tinuous Galerkin Methods, Springer.

Domelevo K., Omnes P. (2005) A finite volume method for theLaplace equation on almost arbitrary two-dimensional grids,ESAIM: Mathematical Modelling and Numerical Analysis 39,(6), 1203-1249.

0 1 2 3 4 5 6

0.001

0.01

0.1

1

Time

Mas

s er

ror

0 1 2 3 4 5 6

0.001

0.01

0.1

1

Time

Mas

s er

ror

= 10−3∋

∋

∋∋

∋

∋

∋

= 10−5

= 10−3 − P1

= 10−3 − P2= 10−4 − P1

= 10−3 − P3

= 10−4

Figure 12

TC3 – mass error for DDFV (solid lines) and SWIP

(dashed lines).


Ern A., Vohralık M. (2010) A posteriori error estimation basedon potential and flux reconstruction for the heat equation,SIAM Journal on Numerical Analysis 48, (1), 198-223.

Eymard R., Guichard C., Herbin R., Masson R. (2012) Vertex-centred discretization of multiphase compositional Darcy flowson general meshes, Computational Geosciences, pp. 1-19.

Fagherazzi S., Furbish D.J., Rasetarinera P., Hussaini M.Y.(2004) Application of the discontinuous spectral Galerkinmethod to groundwater flow, Advances in Water Resources27, (2), 129-140.

Hairer G., Wanner E. (2010) Solving ordinary differential equa-tions II, Springer.

Haverkamp R., Vauclin M., Touma J., Wierenga P.J.,Vachaud G. (1977) A comparison of numerical simulationmodels for one-dimensional infiltration, Soil Science Societyof America Journal 41, (2), 285-294.

Herbin R., Hubert F. (2008) Benchmark on discretizationschemes for anisotropic diffusion problems on general grids.Finite volumes for complex applications V, pp. 659-692, ISTE,London.

Hermeline F. (2000) A finite volume method for the approxi-mation of diffusion operators on distorted meshes, Journal ofComputational Physics 160, (2), 481-499.

Klieber W., Riviere B. (2006) Adaptive simulations of two-phase flow by discontinuous Galerkin methods, ComputerMethods inAppliedMechanics andEngineering 196, (1), 404-419.

Knabner P., Schneid E. (2002) Adaptive hybrid mixed finiteelement discretization of instationary variably saturated flowin porous media, High Performance Scientific and EngineeringComputing 29, 37-44.

Krell S. (2010) Schemas Volumes Finis en mecanique desfluides complexes, PhD thesis, Universite de Provence-Aix-Marseille I.

Lesaint P., Raviart P.A. (1974) On a finite element method forsolving the neutron transport equations, Universite Paris VI.

Manzini G., Ferraris S. (2004) Mass-conservative finite volumemethods on 2D unstructured grids for the Richards’ equation,Advances in Water Resources 27, (12), 1199-1215.

Narasimhan T.N., Witherspoon P.A. (1976) An integratedfinite difference method for analyzing fluid flow in porousmedia, Water Resources Research 12, (1), 57-64.

Simmons J., Landrum B.L., Pinson J.M., Crawford P.B. (1959)Swept areas after breakthrough in vertically fractured five-spotpatterns, Trans. AIME 216, 73.

Sochala P., Ern A., Piperno S. (2009) Mass conservative bdf-discontinuous galerkin/explicit finite volume schemes for cou-pling subsurface and overland flows, Computer Methods inApplied Mechanics and Engineering 198, (27), 2122-2136.

Van Genuchten M.T. (1980) A closed-form equation for pre-dicting the hydraulic conductivity of unsaturated soils, SoilScience Society of America Journal 44, (5), 892-898.

Final manuscript received in May 2013

Published online in November 2013

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