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; Z Ls_ > B´d : ]^BGb=ÏÀ_ > B É L?KM_^B$B�CMB�y�BDLk_�]ed > B�y�B : CEBD_³ñ � NRB�L?bc_^BG_ > B < KMB)deB å KM]^BGCMKML?B : H É LpKE_JBGB�CMB�y�BDLk_ :�<?< H^b ?KMy : _JKMb�L_^b�_ > B#]^b�CMo?_JKMb�L : _ > B�Lª_ > B :�<?< H^b ?KMy : _JKMb�L#B�H^HJb�H * � ù ñ ê ñ � KE] : Ï¥opL@de_^KMbcL/b�Ï�� �hëíýGï [: L@N b�L?B¨y :hÎ_ > B�HJBeÏ¥bcH^B(B : ]^KMC Î NRB É L?B�KM_^] : � : L@N � �� L?b�HJy : ] V

* � � 6 ù ë 1 � ëíñòë¥î ï ê ñ � ë¥î ïJï � 6cî ï 46 û ë iku ï * � � 4� ù ë 1 � 1 ð ñòë¥î ï ê ð ñ � ë�î ï 1 � 6�î ï 46 û ë j�w ï

; Z L _ > B/d : ]JB�b�Ï(_ > B É L?KM_^B�Q�b�CMo?y¨B�]ed > B�y�B [ _ > B :=<?< HJb ÊKMy : _^KMbcL²B�HJH^bcH*d : L�ImB�B� < HJB�]^]JB)N : ] : < KMB)deB å KM]^Bdeb�L?]J_ : Lk_¾Ï¥o?L@de_JKMb�L *��� Ï¥H^bcy&ý _Jb Z 9�åG> B�H^B¨_ > B Q : CEopB(b�Ï *��� b�L : T�KMQ�B�L3_^HJK : L?TcCEB 5 ü<* b�ÏG_ > B y¨B�] >

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70 72 74 76 78 80 82 84 86 88 90

(degree)

Greater triangle angle

1.8

2.0

2.2

2.4

2.6

2.8

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*10-2

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70 72 74 76 78 80 82 84 86 88 90

1.8

2.0

2.2

2.4

2.6

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l KETco?H^B imVrl KML?KM_^B � bcCEopy�BG]�d > BDy�B�I@B >@: QRKEbcH+bcL : CEKMy¨KE_d : ]^B [p< o?HJBsNRKÌËÀop]^KMb�L#d : ]JB [ a�H^bcI?CMB�y&ë i �cï�Íeë i �cï U

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0.0 0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2

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Finite Element in H1 semi-norm Finite Volume in H1 semi-norm

Finite Element in L2 norm Finite Volume in L2 norm

l KMTco?H^B j?Vl KML?KM_^B � b�CMo?y¨B [Rl KML?KM_JB $³CEBDy�B�LR_Gdeb�y <@: HJKE]Jb�L [c< o?HJB(NRKÌËÀo?]JKMb�L�d : ]JB [ a³H^b�IpCEBDy&ë i ��ïíÍ�ë i �cï U

KE](debcL?]^_ : LR_ : L@N�T�KMQcB�L�I Î-V *��4 ù ñòë¥î 4 ï ê ñ 4 U(èG> B : � L?bcH^y&b=Ï+_ > B :=<?< H^b ?KMy : _^KMb�L¨B�H^HJb�H *�� KM]P_ > B�LNRB É L?BhN#I ÎÀV * �� � 6 ù 3-ë �4?�#� ë * ��4 ï � ï 46 û ë j?f ï

� B�NRB É L?B : ä NÊKE]edeH^BD_^B � �� ä L?b�HJy : ] V * �� �� 4� ù¿ë �W ��� 1 * ��4 �� ï ê

* ��4 �� 1 � ï 46 û ë j�v ï

èG> B : � : LmN � �� Lpb�H^y b�Ïr_ > B :�<?< H^b ?KMy : _JKMb�L(B�HJH^bcH : H^B < CMbc_^_^BhN�QR]G_ > B�y�B�] > ]^KMæ�B�KML l KMT�opH^B j?U¼ß ]GB� < B)de_JB)N [ _ > BÉ L?KM_JB+B�CMB�y¨B�LR_ : � L?b�HJy b=Ï@_ > BGBDH^H^bcH] > b å ] : çco : NRH : _^K�d < B�HéÏ¥b�HJy : LmdeB [cåG> KMCEB8_ > B É LpKE_JB+Qcb�CMo?y¨B : � L?bcH^y¿KML@deHJB : ]JB�]CMKML?B : HJC Î(å KM_ > _ > BGy�B�] > ]^KMæ�B�ë¥_ > KM]KM]KML : TcH^B�B�y¨B�LR_ å KE_ > èG> B�bcH^B�y f ï U�èG> B � �� Lpb�H^yÔb�Ï _ > B :�<?< H^b ?KMy : _JKMb�L�B�H^HJb�Hb�Ï�I@bc_ > ]�d > BDy�B�]PI@B >m: Q�B�]GCMKML?B : H^C ÎkUèG> B : � L?b�HJy b�Ï-_ > B :�<?< HJb ?KEy : _JKEbcL�B�H^HJb�H$b�Ï Imb�_ > ]ed > B�y�BD] : HJB < CMb�_J_^B)N QR]_ > B X a \ _^KMy�B$KML l KETco?H^B xpU èG> B;N : _ :

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0 500 1000 1500 2000 2500 3000

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Finite Element Finite Volume

l KMT�o?HJB xpVl KELpKE_JB � b�CMo?y¨B [�l KML?KM_^B�$�CMB�y�BDLk_ X a \ _JKEy¨B�debcy <@: H^KM]^bcL [c< o?H^B(NRKÌË-o?]^KMbcL*d : ]^B [ a�HJb�I?CMB�y2ë i ��ï�Íeë i ��ï U

]^_JH^o@d"_^o?HJB b�Ï_ > B É L?KM_^B(BDCEBDy�B�LR_;deb?NRB(KM]Py�b�HJB�debcy < CMB�#_ >@: L#_ >@: _;b�Ï_ > B É L?KM_JB(Q�bcCEopy�B�d"bÊNÊB�Ï¥b�HP_ > B�]^b�CMo?_JKMb�L#b�Ï_ > B�] : y¨B < HJb�I?CMB�y U èG> KM]GB� < C : KELp]+_ > B(]^bcy�B åG>m: _$CEb å B�H$B ��deKMB�L@d Î b=Ïr_ > B É L?KM_JB�BDCEBDy�B�LR_Gdeb?NRB U

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3*10-2

4*10-2

5*10-2

6*10-2

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0.0 0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2

2*10-2

3*10-2

4*10-2

5*10-2

6*10-2

Finite Element Finite Volume

l KMT�o?HJB � V�l KML?KM_^B � b�CMo?y¨B [l KML?KM_^B"$³CEBDy�B�LR_ deb�y <@: HJKE]Jb�L/bcLªCMb?d : CMC Î HJB É L?BhNôy�B�] > [< o?H^B�NRKÌË-o?]^KMb�L�d : ]^B [ a�HJb�IRÍCMB�y ë i ��ïíÍ�ë i �cï U

l KMT�o?HJB �P] > b å ]_ > BGImB >@: QRKMb�H�b=Ï@_ > B :=<?< HJb ÊKMy : _^KMbcL�BDH^H^bcHb�L _ > B É L?KM_^B$B�CMB�y�BDLk_ : L@N É L?KM_^B$Q�b�CMo?y¨B]�d > BDy�B�] åG> B�L_ > B É H^]^_PCMKELpB : LmN _ > B�C : ]^_;debcCEopy�L b�Ï�_ > B�y¨B�] >#: H^B�HJB É L?B)Nãë¥]^BDB l KETco?H^B g ï U ��> B�L#_ > B�HJB É L?BDy�B�LR_G]^_JB < _^B�LmNR];_Jbæ�B�HJb [ _ > B(_ å b ]�d > B�y¨B�];ImB >@: QcB�]JKMy�KMC : H^C Îs: ]G]^B�BDL�b�L l KMT�o?HJB � Ul KMT�o?HJB ��] > b å ] _ > B < HJB)deKM]^KMb�Lôb=Ï�_ > B :=<?< HJb ÊKMy : _^KMbcL�b�Ï�_ > B �@o�?B�]1Ï¥b�H _ > B É L?KM_^B#B�CMB�y�BDLk_ : LmN É L?KM_JB�Qcb�CMo?y¨B]�d > BDy�B å KM_ > H^BD] < B)de_;_Jbs_ > B;y¨B�] > ]^KMæ�B U�Z _$KM]´d"CEB : H8Ï¥H^bcy _ > B�H^B�]Jo?CM_^]G_ >@: _$_ > B É L?KM_^BPQ�bcCEopy�B$]�d > B�y¨B�TcKMQ�B�] : y�bcH^B< HJB)deKM]^B :=<?< HJb ÊKMy : _^KMbcLsb=ϳ_ > B��@o� U�èG> KM]Gy :hÎ ImB�b=ÏKEy < bcH^_ : L@deBPKML�deB�HJ_ : KEL3deb�o < CEBhN < H^b�IpCEBDy�]Pë¥]^BDB(B U T U�� �&¥ï U

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0.0*100

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Finite Element Finite Volume

l KETco?H^B�� Vl KML?KM_JB � bcCEopy�B [Rl KELpKE_JBz$�CMB�y¨B�LR_;deb�y <@: HJKM]^b�L bcL �@o�?B�] [p< o?HJBsNRKÌËÀop]^KMb�L#d : ]JB [ a�HJb�I?CMB�y2ë i ��ïíÍ�ë i ��ï U

��� ���"�.+S)Q�"����&�� ����&(� #��"-��/ �����(���0/ ��#��0+(������� �0���"��&(�%��� ����&(� #��"-��/ � ���(�.�%/ � ���� ���B� �������"� �X�!���"#��0�'&(���"� ).+(�",.-��%/

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6 @ ZRë�ñ-ëBA � û A � ïJð ë A � û A � ï^ï ê � ñòë A � û A � ï ù ë x A � � ê v ï * P 6 û ë A � û A � ïü�ý û ë j�g ïñòë A � û A � ï ù A � � * P 6 û ë A � û A � ï$ü �mý û ë j�i ï ë A � û A � ï ù A � � ó A � û ë A � û A � ï8ü�ý û ë j�j ï

åG> B�HJBsýÖKE] :�<@: H : CMCMB�CMb�TcH : y NÊB É L?B)N*b�L l KETco?H^B gp[RåG> K�d > KM]GNRKM]�deHJB�_^KMæ�B)N å KM_ >#: L#KMH^H^BDT�o?C : HG_JH^K : LpT�o?C : H$y�BD] > U

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Finite Volume Weighted Finite Volume

l KMT�o?HJB u?V�l KML?KM_^B � b�CMo?y�B [ � BDKET > _^B)N l KML?KM_^B � b�CMo?y¨B´debcy <@: H^KM]^bcL [ debcLRQ�B)de_JKMb�LRÍ^NRKÌËÀop]^KMb�L�d : ]^B [ a³H^b�IpCEBDy ë jcg ï�Íeë jcj ï Ul KMT�o?HJB u ] > b å ]P_ > B :�<?< H^b ?KMy : _JKMb�L�BDH^H^bcHPÏ¥b�HP_ > B É L?KM_^B�Q�b�CMo?y¨B : L@N å B�KMT > _JB)N É L?KM_JBsQcb�CMo?y�B�NRKM]�deHJB�_^KMæ : _^KMb�Lp](b�Ï< HJb�I?CMB�y ë jcg ïíÍ�ë j�j ï U�èG> B :=<?< HJb ÊKMy : _^KMbcL(B�H^HJb�H$b�Ï�_ > B�deb�LRQ�Bhde_^KMb�L¨_^B�HJy�KM];ë�_ > BDb�H^BD_^K�d : CMC Î ïb=Ï b�HeNRB�H fGå KM_ > H^B�] < Bhde__^b _ > B;y�BD] > ]^KMæ�B�Ï¥b�H$Imb�_ > y¨B�_ > bÊNÊ] U�èG> B�NRKÌËÀo?]JKMb�L _^BDH^y�KM]ImB�_^_JB�H :=<?< HJb ÊKMy : _^BhN(I Î _ > B å B�KMT > _JB)N É L?KM_^B;Qcb�CMo?y¨B

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ñ � ëBA � û A � û � ï-ó 6 @ ZRë ø ë�ñ@ïJð��-ë A � û A � ï^ï ù vr� ó v ë A � ó9A � ï"ë A � A � ó � � ï û ëJë A � û A � ï û�� ï8ü�ý�� wpûe5��Mû ë j�x ïñòë A � û A � û � ïþù A � A � ó � � û ëJë A � û A � ï û�� ï8ü��@ý��� w?ûx5��Mû ë j �cïñ-ëBA � û A � û�w ïþù A � A � û ëBA � û A � ï$ü�ý û ë j �cï

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ø ëíñmï5ù ñ � û ë j�u ï�-ëBA � û A � ïúù A � ó=A � û ëBA � û A � ï8ü²ý û ë x�w ï

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Finite Volume Weighted Finite Volume

l KMT�o?HJB f)w?Vl KML?KM_^BPQ�bcCEopy�B [cå BDKET > _^B)N É L?KM_JB�Qcb�CMo?y¨B;deb�y <@: HJKE]Jb�L [c>RÎR< B�HJI@bcCMK°d�d : ]^B [ a�H^bcI?CMB�y2ë j�x ï�Íeë j ��ï U

l KMT�o?HJB fhw ] > b å ];_ > B :�<?< H^b ?KMy : _JKMb�L B�H^HJb�H å KM_ > H^BD] < B)de_�_^b _ > B(y�BD] > ]JKMæ�B�Ï¥bcH+Imb�_ >�É L?KM_^B�Q�bcCEopy�B : LmN å B�KMT > _JB)NÉ L?KM_JB�Q�bcCMo?y�B(]ed > B�y�B U _ > B¨I@B >m: QkKMbco?Hsb=Ï´_ > B�BDH^H^bcH(Ï�b�H(Imb�_ > ]�d > BDy�B�] KM] : T : KML²b=Ï;b�HeNRB�HI3 ��� � U�à bc_^B¨_ >@: _(_ > B_ > B�b�HJB�_^K�d : C B�HJH^b�HPB�]^_JKMy : _JB�]GT�KMQcB : L#bcH�NRB�HGb=Ï 3 �����èG> BP]�d > B�y¨B�] l �¿: L@N � ��l3: HJB;]JKEy¨KMC : H åG> BDL�KMy < CMB�y¨B�LR_^B)N�b�L _ > B;bcL?BeÍ^NRKMy�B�Lp]^KMb�L : C t o?H^TcB�H^]GB)çco : _^KMb�L : KML Ï : de_ [_ > B β: HJB ]^KMy�KMC : H$KÌÏ : o?L?KÌÏ¥bcy4y¨B�] > KM]�o?]JB)N U èG> B Î�å B�HJB I@bc_ > KEy < CMB�y�BDLk_eN Ï¥bcH :�9 KMB�y : L?L < HJb�I?CMB�y Î KEBDC°NÊKELpT :] > bÊdx� < H^b <@: T : _^KMbcL Urt b�_ > ]�d > B�y¨B�];] > b å : L :=<?< H^b ?KMy : _^KMb�L B�HJH^b�HPb�Ï�b�H�NÊB�Hz3 46 UOPL?B y :hÎ�: N?N : b�L?B NRKMy�B�L?]JKMb�L : C �+: L²SòB�B�H < H^b?deB)NRopH^B Ï¥bcH;_ > B�deb�y < o?_ : _^KMb�L¨b�Ï+_ > B��@o�?B�](KML�_ > B6NÊKEHJB)de_JKEbcL

b�HJ_ > bcT�bcL : C?_Jbs_ > B;BhNRT�B�]Gb=Ï _ > BPy�B�] > U¼èG> KM]+y¨B�_ > b?N åG: ]$]^o@dDdeB�]�Ï�o?CMC Î op]^B)N¨KML �Mf @Ï¥bcH+_ > Bz$³o?CMB�H+Bhç�o : _^KMbcL?] U ��> B�LKMy < CMB�y¨B�LR_^B)N�Ï�b�H$_ > B t o?HJT�B�HJ];B)çco : _^KMb�L [?: NÊB)deH^B : ]^B b�Ï�_ > B�B�H^HJb�HGb�Ïo < _Jb jcw < B�HedeB�LR_;KM]GbcI?_ : KELpB)N U

f�i

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��� � � � ��� ��� � ���0-�-��)�&(���'� � A�)��!+(,��"-�B� �!�"�.)�-�%� �!A�� &(�%/X b�Lp]^K�NRB�HG_ > BPÏ¥b�CMCMb å KML?T�B�CMCEK < _JK�dJÍ >RÎR< B�H^Imb�CMK�d;debco < CMB)N¨] Î ]J_^B�y [RåG> K�d > y�b?NRB�CM]$_ > B��@b å b�ÏrbcKMC : L@N å´: _^B�HG_ > H^bco?T >#:< b�H^bco?]Gy�BhNRKMo?y [ KML@d"CEomNRKML?Ts_ > B�T�H : QkKM_ Î BeË-B)de_ V

���eë�î û�� ï ê NRKMQmë �=ë¥îû � ï

��� ëíð��òë¥î û � ï ê�� ���ÿRï^ïúù w?û ë¥î û � ï8ü*ý��� w?û * �Eû ë x?f ï

ê ���eë�î û�� ï ê NRKMQmëf ê �=ë¥î û � ï

��� ëíð��òë¥î û � ï ê� ��ÿRï^ïúù w?û ë¥î û � ï8ü*ý��� w?û * �Eû ë x�v ï�=ë¥î û � ïúù � � ë¥î û � ï û ë¥î û � ï8ü �mý��� w?û * �Mû ë x�g ï�=ë¥î ûew ïþù � � ë¥î ï û î3ü�ý û ë x�i ï�-ë�î û�� ïþù � � ë¥î û � ï û ë¥î û � ï�ü �mý��� wpû * �Mû ë x�j ïñòë¥î ûew ïþù ñ � ë¥î ï û î3ü*ý û ë x�x ï

Homogeneous porous media

1 and 2 are reference cells

Oil

Water

x

y

2

1

l KMTco?H^B f�fcVrY B�] > NÊB É L?KM_^KMbcL�Ï�b�HGdeb�y <@: HJKE]Jb�L?]åG> B�HJBsýÖKE] : <@: H : CMCMB�CMb�TcH : y NRB É L?B)N#bcL l KMT�opH^B f�fc[ �ÿ KM]G_ > B�TcH : QRKM_ Î�: d�d"B�CMB�H : _^KMb�L#Q�Bhde_^bcH : LmN�� � [ � � ë¥HJB�] < B)de_JKMQ�BQRKM]�debc]^KM_^KMB�]�b�Ï åG: _^BDH : L@N�b�KMC�ï [ � � : L@N � � ë¥HJB�] < B)de_JKMQ�B�NÊB�L?]^KM_JKEBD](b�Ï åG: _JB�H : L@N�b�KMC�ï : H^B T�KMQ�BDLãdebcL?]^_ : LR_^] U Z L²_ > KM]d : ]^B [ ëíì � ó/ì � ï"ë��hïù � �� ó � � � �� [ ]Jb�ë�ì � ó/ì � ï$NRB < B�L@NR]Pb�L"� Uà bc_^B�_ >m: _+�;HJB < HJB�]^B�LR_J]�_ > B å´: _^B�H <?>@: ]^B�] : _^o?H : _JKEbcL: LmN8�#_ > B�_ å b <p>@: ]JB�] < HJB�]^]Jo?H^B UGèG> B(NRb�y : KML�ýÖKE];NÊKE]edeH^BD_^KMæ�B)N å KM_ >�: L¨KEHJH^B�Tco?C : HG_JH^K : L?Tco?C : H$y�B�] > ë¥]^BDB l KET fcf ï UèG> B ä B� : de_ ä ]^bcCMo?_^KMb�L åG: ]rd"b�y < o?_^B)N�o?]JKELpT : Q�B�H Î�É L?BGy¨B�] > : L@N :�l o��"KE_J]^o(]Jo < B�H�debcy < op_^B�H [ o?]JKML?T�I@bc_ > ]�d > B�y¨B�] [: LmN�_ > B�H^B�]Jo?CM_^] : H^B�B)çco : C-o < _Jb _ > B�y : d > KML?B < HJB)deKM]^KMbcL UZ L l KMT�opH^B�] f)vp[ HJB�]^o?CM_J]Ï¥bcH+_ > B;o <kå KML@N d > bcK�deBGÏ¥bcH+Imb�_ > ]�d > B�y¨B�] : H^B < HJB�]^BDLk_JB)N U8èG> B;bcH�NRB�HGb=Ï _ > B :�<?< H^b ?KMy : _JKMb�LKM]z3 ��� � KML�Imb�_ > d : ]JB�] [ I?o?_P_ > B É L?KM_^B Q�bcCMo?y�BP]�d > B�y¨BsKM];]Jb�y¨B åG>@: _Gy¨b�HJB < H^BhdeKM]^B U l KMTco?H^B fhg ] > b å ]�_ > B H^B�]Jo?CM_^]Ï¥bcH : �+: LãSòB�B�H :=<?< HJb ÊKMy : _^KMbcL�b=Ï+_ > B�@o�ôë : ](NRBD]�deHJKEImB)N KML²_ > B < HJB)deB)NRKML?T <@: H : T�H :�<?> KML²Imb�_ > ]ed > B�y�BD] U¨èG> B:�<p< H^b ?KMy : _^KMb�L�BDH^H^bcHsKM] y o@d > CMb å B�H¾Ï¥b�H I@bc_ > ]�d > B�y¨B�] : KE_ d : LªImB�L?bc_^K�deB)N _ >@: _s_ > B#KEy < H^b�QcB�y B�LR_�KM](I@BD]^_ Ï¥bcH_ > B å B�KMT > _JB)N É LpKE_JB�Q�b�CMo?y¨BG]�d > B�y¨B [RåG> K�d > KM]$KEL : TcH^B�B�y¨B�LR_ å KM_ > _ > B;Ï : de_G_ >@: _$LRo?y�B�HJK�d : C-NRKÌËÀop]^KMb�L¨b�Ï å B�KMT > _JB)NÉ L?KM_JB�Qcb�CMo?y¨BGKE]GC : H^T�BDHG_ >@: L�_ >@: _$b�Ï É LpKE_JB�Q�b�CMo?y¨B U� B$L?b åª< HJB�]^BDLk_ : ]^_Jo@N Î b�Ï-_ > BG] : _^opH : _JKMb�L å KE_ > HJB�] < B)de_$_^b�_ > BG_JKEy¨B U'� B+]J_ : H^_ å KM_ > : NRKM]�debcLR_^KMLRo?b�o?]8] : _Jo?H : _^KMb�L Vå B+_ : ��B=�;ù f KML(_ > B+] >@: NRBhN deB�CMCM] b�Ï l KMTco?H^B f�f$: L@N��Pù w B�CM]^B åG> B�H^B U�� B] >m: CMCR]^_^omN Î _ > B+Q : CEopB�] b=Ï@_ > B$] : _^o?H : _JKEbcLKMLãd"B�CMCE] f : LmN v NRB É L?B)N/KML l KETco?H^B f�fcUsl KMT�o?HJB�] fDi*: L@N f)j ] > b å _ >@: _�_ > B HJB�]^o?CM_J]sTcKEQcB�L�I Î Imb�_ > ]�d > B�y¨B�] : HJBdebcy <@: H : IpCEB [k: L@N¨_ >@: _G_ > B �+: L*SòB�B�H :=<?< H^b ?KMy : _^KMb�L H^BhNRo@deB�]�_ > B(LRo?y�BDH^K�d : C-NRKÌË-o?]^KMb�L U

f)j

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0.0 0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2

Mesh step

0.000.020.040.060.080.100.120.140.160.180.200.220.240.260.280.300.320.340.360.380.40

App

roxi

mat

ion

erro

r in

L2

norm

0.0 0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2

0.00.020.040.060.080.10.120.140.160.180.20.220.240.260.280.30.320.340.360.380.4

Finite Volume Weighted Finite Volume

l KMT�o?HJB fhv?V�X b�y <@: HJKE]Jb�L å KM_ >�: L�o < ]J_^HJB : y ]ed > B�y�B [ d"b�o < CMB)N*] Î ]^_JB�y d : ]^B [ a³H^b�IpCEBDy ë xpf ï�Íeë xcx ï U

0.0 0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2

Mesh step

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

App

roxi

mat

ion

erro

r in

L2

norm

0.0 0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2

0.0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Finite Volume Weighted Finite Volume

l KMT�o?HJB f)g?VX bcy <@: H^KM]^bcL å KM_ >�: �$: LRÍ^S-B�BDHG]�d > B�y¨B [ debco < CMB)N¨] Î ]J_^B�y&d : ]^B [ a�H^bcI?CMB�y2ë x?f ï�Íeë xcx ï U

l KMT�o?HJB fhx ] > b å ];_ >@: _G_ > B É L?KM_^B�Qcb�CMo?y�BG]ed > B�y�B�KM];debcy < o?_ : _JKEbcL : CMC Î d > B :=< BDHGÏ¥bcH : TcKEQcB�L < H^BhdeKM]^KMb�L Uß CMb?d : C H^B É L?B�y¨B�LR_;b�Ï_ > B(y�BD] > ë¥KML : ]JKEy¨KMC : H : ]�NRB�]edeH^KMImB)N�bcL l KMTco?H^B g ïGKM]G_ > B�L < B�H�Ï¥bcH^y¨B)N : _ > B É L?KM_^B(Qcb�CMo?y¨B]�d > BDy�B�H^B�y : KML?]$y�bcH^B < H^B)deKM]JB(KML�_ >@: _;d : ]^B ë�]^B�B l KETco?H^B f ��ï UZ L l KETco?H^B f �(_ > B(B�HJH^b�HPb�L¨_ > B��mo�?B�]GKM]G] > b å L U�ß T : KEL¨_ > B Î*: H^B�ImB�_^_JB�H :=<?< H^b ?KMy : _^KMb�L å KM_ >¨É L?KM_JB�Qcb�CMo?y¨B Ul KMT�o?HJB f)u(: L@N vcw ] > b å _ > B�NÊKE]J_^HJKEIpo?_^KMb�L b�Ï-_ > B :�<p< H^b ?KMy : _^KMb�L�B�HJH^b�H : _ É L : Cp_^KMy�B 5 ù 5 � >JI W�� Ï�b�HImb�_ > ]�d > B�y¨B�] UZ _Gd : L¨I@BPH^B�y : He�cB)N _ >m: _$_ > BPB�H^HJb�H8Ï¥b�H$_ > B É L?KM_^BPQ�bcCEopy�B$]�d > B�y¨BGKE] > KMT > B�H+LpB : H$_ > BPI@bco?L@N : H^KMB�] åG> KMCMBGKE_8KE]$y�bcH^Bo?L?KÌÏ¥bcH^y¨C Î NRKM]^_^HJKMI?o?_^BhN#KML�_ > Bsd : ]JB(b�Ï å BDKET > _^B)N É L?KM_JB�Qcb�CMo?y¨B U

f)x

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-8 -7 -6 -5 -4 -3 -2 -1 0

Time in million years

0

10

20

30

40

Oil

satu

ratio

n in

per

cent

-8 -7 -6 -5 -4 -3 -2 -1 0

0

10

20

30

40

Four Point Finite Volume for the cell 2 Weighted Finite Volume for the cell 2

Four Point Finite Volume for the cell 1 Weighted Finite Volume for the cell 1

l KMT�o?HJB f�i@V # : _^o?H : _^KMbcL*d"b�y <m: H^KM]Jb�L å KM_ >�: L¨o < ]J_^H^B : y ]�d > BDy�B [ debco < CMB)N#] Î ]^_^BDy d : ]^B [ a�HJb�I?CMB�y ë x?f ï�Íeë xcx ï U

-8 -7 -6 -5 -4 -3 -2 -1 0

Time in million years

0

10

20

30

40

50

Oil

satu

ratio

n in

per

cent

-8 -7 -6 -5 -4 -3 -2 -1 0

0

10

20

30

40

50

Four Point Finite Volume for the cell 2 Weighted Finite Volume for the cell 2

Four Point Finite Volume for the cell 1 Weighted Finite Volume for the cell 1

l KMT�opH^B f)jpV # : _Jo?H : _^KMb�L#deb�y <@: HJKE]Jb�L å KE_ >�: �+: L�SòB�B�H;]ed > B�y�B [ debco < CMB)N¨] Î ]^_JB�y&d : ]^B [ a�H^bcI?CMB�y&ë x?f ï�Íeë x�x ï U

f �

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0.0*100 2.0*103 4.0*103 6.0*103 8.0*103 1.0*104 1.2*104

(s)

CPU time

0.0

0.1

0.2

0.3

0.4

App

roxi

mat

ion

erro

r in

L2

norm

0.0*100 2.0*103 4.0*103 6.0*103 8.0*103 1.0*104 1.2*104

0.0

0.1

0.2

0.3

0.4

Weighted Finite Volume Finite Volume

l KMT�o?HJB f)xpVãl KML?KM_^B � bcCMo?y�B [�� B�KMT > _^BhN l KELpKE_JB � b�CMo?y¨B X a \ _^KMy¨B�debcy <@: H^KM]^bcL [ deb�o < CEBhNô] Î ]^_JB�y�d : ]^B [ a�HJb�IRÍCMB�y ë x?f ïíÍ;ë xcx ï U

0.0 0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2

Mesh step

1.0*10-1

1.5*10-1

2.0*10-1

2.5*10-1

3.0*10-1

3.5*10-1

4.0*10-1

App

roxi

mat

ion

erro

r in

L2

norm

0.0 0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2

1.0*10-1

1.5*10-1

2.0*10-1

2.5*10-1

3.0*10-1

3.5*10-1

4.0*10-1

Weighted Finite Volume Finite Volume

l KMT�o?HJB f � V+l KML?KM_^B�Q�bcCEopy�B [cå BDKET > _^B)N É L?KM_^B(Qcb�CMo?y�B�deb�y <@: HJKM]^b�L¨b�L#CMb?d : CMC Î H^B É L?B)N�y�B�] > [ debco < CMB)N#] Î ]^_^BDy&d : ]JB [a�HJb�I?CMB�y2ë xpf ï�Íeë xcx ï U

f �

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0.0 0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2

Mesh step

0.0*100

2.5*10-2

5.0*10-2

7.5*10-2

1.0*10-1

App

roxi

mat

ion

erro

r

0.0 0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2

0.0*100

2.5*10-2

5.0*10-2

7.5*10-2

1.0*10-1

Weighted Finite Volume Finite Volume

l KMT�o?HJB f � Vl KML?KM_^BPQ�b�CMo?y¨B [cå B�KMT > _JB)N É L?KM_^B�Qcb�CMo?y�B;d"b�y <m: H^KM]Jb�L b�L �@o ÊBD] [ d"b�o < CMB)N¨] Î ]J_^B�y&d : ]^B [ a³H^b�IpCEBDy ë xpf ï�Íë xcx ï U

0.0 0.2 0.4 0.6 0.8 1.0

X

0.0

0.2

0.4

0.6

0.8

1.0

Y

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Below 1.0*10-3

1.0*10-3 - 1.4*10-3

1.4*10-3 - 1.8*10-3

1.8*10-3 - 2.2*10-3

2.2*10-3 - 2.6*10-3

2.6*10-3 - 3.0*10-3

3.0*10-3 - 3.4*10-3

3.4*10-3 - 3.9*10-3

3.9*10-3 - 4.3*10-3

4.3*10-3 - 4.7*10-3

4.7*10-3 - 5.0*10-3

Above 5.0*10-3

Approximation error

l KMT�o?HJB f)u?Vòß;<?< HJb ?KEy : _JKEbcL�B�H^HJb�H�Ï¥bcH¼Ï¥b�o?H < b�KMLR_ É L?KM_^BGQcb�CMo?y�B8]�d > B�y¨B [ debco < CMB)N ] Î ]J_^B�y�d : ]^B [ a�HJb�I?CMB�y ë xpf ï�Íeë xcx ï U

f)u

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0.0 0.2 0.4 0.6 0.8 1.0

X

0.0

0.2

0.4

0.6

0.8

1.0

Y

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Below 5.0*10-4

5.0*10-4 - 4.5*10-3

4.5*10-3 - 8.6*10-3

8.6*10-3 - 1.3*10-2

1.3*10-2 - 1.7*10-2

1.7*10-2 - 2.1*10-2

2.1*10-2 - 2.5*10-2

2.5*10-2 - 2.9*10-2

2.9*10-2 - 3.3*10-2

3.3*10-2 - 3.7*10-2

3.7*10-2 - 4.0*10-2

Above 4.0*10-2

Approximation error

l KMT�o?HJB v�wpV ßP<?< HJb ÊKMy : _^KMbcLsB�HJH^bcH$Ï¥b�H å B�KMT > _JB)N É L?KM_^BPQ�b�CMo?y¨B+]ed > B�y�B [ deb�o < CEBhN ] Î ]J_^B�y d : ]JB [ a³H^b�IpCEBDy ë xpf ï�Íeë xcx ï U

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