28
Modeling wells in porous media flows Pierre Fabrie, Thierry Gallouet To cite this version: Pierre Fabrie, Thierry Gallouet. Modeling wells in porous media flows. Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2000, volume 10, numero 5, pp.673-709. <hal-00003333> HAL Id: hal-00003333 https://hal.archives-ouvertes.fr/hal-00003333 Submitted on 20 Nov 2004 HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destin´ ee au d´ epˆ ot et ` a la diffusion de documents scientifiques de niveau recherche, publi´ es ou non, ´ emanant des ´ etablissements d’enseignement et de recherche fran¸cais ou ´ etrangers, des laboratoires publics ou priv´ es.

Modeling wells in porous media flows - CORE · Modeling wells in porous media ows Pierre Fabrie, Thierry Gallouet To cite this version: Pierre Fabrie, Thierry Gallouet. Modeling wells

Embed Size (px)

Citation preview

Page 1: Modeling wells in porous media flows - CORE · Modeling wells in porous media ows Pierre Fabrie, Thierry Gallouet To cite this version: Pierre Fabrie, Thierry Gallouet. Modeling wells

Modeling wells in porous media flows

Pierre Fabrie, Thierry Gallouet

To cite this version:

Pierre Fabrie, Thierry Gallouet. Modeling wells in porous media flows. Mathematical Modelsand Methods in Applied Sciences, World Scientific Publishing, 2000, volume 10, numero 5,pp.673-709. <hal-00003333>

HAL Id: hal-00003333

https://hal.archives-ouvertes.fr/hal-00003333

Submitted on 20 Nov 2004

HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, estdestinee au depot et a la diffusion de documentsscientifiques de niveau recherche, publies ou non,emanant des etablissements d’enseignement et derecherche francais ou etrangers, des laboratoirespublics ou prives.

Page 2: Modeling wells in porous media flows - CORE · Modeling wells in porous media ows Pierre Fabrie, Thierry Gallouet To cite this version: Pierre Fabrie, Thierry Gallouet. Modeling wells

������������� ������������������������������ �!"�����

#%$'&)(+*',.-0/21�(+3'4�5�$'6�(270708:9<;/>=@?

ACBEDGFHGI:J2FLK0MONQP:RTSVUXW>U'Y.Z[�\%YVU:Z.]^�Y�NQP_YCYG`�RTSaNaYbMXcGY�]>de\%YGWf�Sa]gih_NQRT]M_Sjdk]>Z�lmWnNQP_Y@lmW�NQRTcGWg�lm]2o:YbgTSj]d�l�RTSacbRip:gTY�W>MXoRTlml�RTSac.RipqgTYsr_]\ONQP:Z.]>hXt>P�U']Z.]>hXS%lmYGo2Rih_lCu%vwMxRTl�U']>Z.NaWM:N�o+Riy�c.hqgTN{zVc@]�lmYGS<d0Z.]nl|NQPXY}lx]2oqYbgiRT~@W�NQRT]>M�]d)NQP_Y�\%YbgigTSG[X\%PqRTcaPo:]2YGSwMX]�N�W>gigT]\�hXS�Na]�hXSaY�cbgTWnSaSQRTcGW>g�^WZaRTW�NQRT]>MXW>g�d�]Z.l�h:gTW�NQRT]M_Ss]d�NQP_YsYG��hXW�NQRT]M_SGu

�sYGz2\%]Z.oqSG����]>M:giRiM_YGWZsU_WZ.W>p']giRTc�YG��h_WnNQRT]>M)[_r_]\�NQPqZ.]h_tP�U']>Z.]h_SslxYGo+RTW:[_YG`�RTSaNaY.M_cGYx]>d<Sa]gih_NQRT]M_SG[�lmYGW�SQh:Z.Yso:W�NaWv}����SQhqp:�aY@cGN}cbgTWnSaSQRi�_c@W�NQRT]>M)���2�>���n�+[X�2�>�s�n�q[��>�2�2�2�

� � �������������}�����

5s�'-0�j�)(2�E/,�-0��4'/�q8:=b/4 =.8�=.�'/�¡�(2=b�'/¡�(2=.-0¢n(+7�(23)(+7¤£_�b-¤�V82¥x�b8:¡�/�¡�8_4'/70��8+¥s¦)82§s�C-03O�E8:,b8:9'��¡�/4'-�('$�5s�'/>�./¡�8_4'/>7¤��(+,./�=.�¨8:�./�9'�./>4O¥{8q,x,b/�./>,.�:8q-0,C�b-¤¡�9¨7k(+=.-08:3�© �s-03O�E/=.,b8:70/9'¡ª/3¨«:-03'//,b-03'«"�./>/­¬i®�¯.°�¬ ±2¯.°�¬i²³¯´°w¬ µ³¯.$�¶a3O=.�'/>�./�',b8:*'70/¡��n°�§�/>7¤70���)(��:/}=.8m*E/�¢8q3_�:/3'-0/3_=b7¤£V¡�8_4'/70/4�$·¶a3��)(2,b=.-0¢9'7�(+,n°2�b-¤3¨¢/s=.�'/}4'-�(2¡�/=./>,�82¥<(�§}/707<¸{(+*)8q9'=�¹�º�¢>¡�»-0���q/,.£��.¡�(2707)¢>8:¡��E(2,./>4�=.8j=.�¨/x70/3¨«:=.��8+¥�=.�¨/x,b/�./>,.�:8q-0,�8:,}=.8�=b�'/m¡�/�b���b-¤¼>/x8+¥½(�¾b,./n(+�.8:3E(2*'70/¿j4'-0�.¢,b/=.-0¼n(+=.-08:3¸.¹�º�=b8�¹�º:º+¡­$.$�$.»>°_8:3'/m�)(+��=b8C¢8q3'�.-04'/>,s=.�'/�(2¢>=.-08:3�82¥�(V§�/707�(+���b8:¡�/s�b�)(2=b-�(27s¾.¡�/n(2�b9',./>�.¿�¸�-03'�.=b/n(24�82¥%¢7�(2�.�b-0¢n(27¥{9'3¨¢=.-08:3¨�@»m§s�'8q�./��.9'�'�E8:,b=.�j�.�',b-03'À­=b8"(��)8q-03X=�¸k¥{8:,�=a§}8"4'-0¡�/3'�.-08q3'3)(27�¡�8_4'/70�G»�8q,�=.8�(��b/«:¡�/3_=�¸0¥Q8:,m=.�',b//4'-0¡�/>3'�.-08:3¨3)(27E¡�8_4'/70�G»$¶a3j=.�'-0���E(2�E/,n°q/Á_-0�.=./>3'¢/�8+¥)§}/n(2Àm�.8q709'=.-08:3¨�<¥{8q,�¡�-0�.¢-0*'70/�(23'4j-0¡�¡�-0�.¢>-¤*¨7¤/·¦)82§s��(2,b/��',b82�:/3�$�Â�¢=.9)(+707¤£_°�9'�.-03'«�=.�'/à ¢G�)(+9'4'/,�Ä)Á_/4j�)8q-03X=�=.�'/>8:,./>¡­°�8:3'/}�',b82�:/�%/Á_-0�.=b/3'¢/s8+¥)§�/�(2Àm�.8q7¤9¨=.-08:3m¥Q8:,�¾b,./«q9'7�(2,b-¤¼>/4�§}/707¤�b¿)$�5s�'/3j=.�',b8:9'«q��.8q¡�/�/>�.=.-0¡�(2=./>��8:3�=.�¨/�./��b8:709'=.-08q3'�n°E8:3¨/C�',b82�:/�V=.�'/j«:/3¨/,@(+7�¢n(2�b/:°��)(2�b�.-03'«�=b8�=.�'/j70-¤¡�-0=�8q3­=b�'/C�b8:709'=b-¤8q3'��8+¥,./>«:9'7�(2,b-0¼/4�/Å_9)(2=b-08:3'��$

Æ Ç ���CÈ���É�Ê�� j���Ë�Ê ���Ì�����w���G���

Í�ÎbÏ ÐÒÑsÓ�Ô�Õ�Ö�×'Ñ�Ø�Õ.Ô�ÙÂÚÄ),b�.=j*)(2�b-0¢�¡�8_4'/>7�-0��4¨/�.¢,b-0*)/4 -03O=.�¨-¤�V�./¢=b-08:3�$�Ûs8:9¨«:�'70£"�b�)/n(+À_-¤3¨«)°�-¤=V¢8:3'�b-0�.=.�j-03�(��)(+,@(2*E8:70-0¢j/Å_9)(2=b-¤8q3¸�-03O=b�'/�9'3¨ÀX3¨82§s3�Ü)»m�E(��X-03'«�( ¾.¡�/n(2�b9',./¿�¸�-03'�.=b/n(24O8+¥x(�¢7�(2�b�.-0¢n(+7�¥{9¨3'¢=.-08q3)»C(+�C(��b8:9',b¢/�=./,b¡Ý(23'4O§s-0=b��(3'8q3'70-¤3¨/n(2,s¢8q3_�:/¢=b-08:3"=b/,.¡�-03_�:8q7¤�_-03'«C(��:/708_¢-0=a£�ÄE/704�«q-0�:/3"*_£�=.�'/��b8:709'=b-¤8q3�82¥�(23�/7070-0�'=.-0¢m/Å_9)(2=b-08:3�¸Þ8:3�=.�'/9'3'À_3'82§s3�ß"§s-0=.�­(�«:-0�:/>3­Ü)»�§s-0=.�­(+7¤�b8�(�¡�/n(2�b9',./C(+�m�.8:9¨,.¢/j=./,b¡­$�5%8�¢8:¡��'70/=./�=b�'/j�',.8q*'70/¡­°_�b8:¡�/V-¤3¨-¤=b-�(27(23¨4C*E8:9'3¨4)(2,b£C¢8q3'4'-0=.-08q3'��(2,./}«:-0�q/3�$ à 9'¢G��¡�8_4'/70�%(2�'�E/n(2,w-¤3j=.�¨/s¡�8_4'/70-0¼n(2=b-08:3m82¥E¦)9'-04V¦)82§s�%=.�',b8:9'«q�C�E8:,b8:9'�¡�/>4'-�('$�à�8q,./�¢8q¡��'70/=b/�¡�8X4¨/70��(2,./m4'/>�.¢,b-¤*E/4��'/,b/n(�¥{=b/,�¸ à />¢=.-08:3�á'$iá:»}*'9¨=s=.�'/�¡�(2-03�4'-ãâ�¢9'70=b-¤/>��(2,./V�',./�b/3_=-03�=b�'/�*)(+�.-0¢m¡�8_4'/7%¸�-03��)(2,b=.-0¢9'7�(+,n°_=.�¨/�8_¢¢9',b/3'¢/j82¥%¡�/n(+�.9',b/��(2�s�b8:9',b¢/C=b/,.¡��@»>$5s�'/"9¨3'À_3'82§s3'��82¥m=.�'-0��¡�8X4¨/7s�',.8q*'70/¡�(2,./�ßä¸Q(��',./�b�.9',b/O-03�,./�(270-0�.=.-0¢�¡�8_4'/>7¤�G»°}Üä¸{(�¢8q3'¢/3_=b,@(2=b-08:3å8q,�(�@(+=.9',G(2=b-¤8q3)»�(+3'4�æ�¸{(xÄ)70=.,@(+=.-08:3j�:/>7¤8_¢-0=a£'»$�5s�'/s/Å_9)(+=.-08:3¨���@(+=.-0�aÄ)/>4�*_£C=b�'/�b/�9'3'À_3'82§s3¨��(2,b/�=.�¨/s¥{8q7¤7082§s-03'«Eç_¥{8q,¸{è�é´ê@»�ë"ì|íî¸Qº'ébï�»°

Ü'ðG¸{è�é´ê@»<ñO4'-0��¸ÞæÉò�¸{ÜE».»>¸{è�éaê@»½óO4'-0��¸Qô�¸�æ�»bõ�Ü)»>¸{è�éaê@»Eñ�Ü�ö2¸{è�é´ê@»a÷�¸{èE»·óùø<úE¸Qè�é´ê@»a÷�¸Qè)»%ûüº'é ¸{á'$0¹�»

æ�¸Qè�é´ê@»)ñ�ýj¸{è�éaênéGÜ�¸Qè�é´ê@».»·¸¤õmß�¸{è�éaê@»·óOþ�¸{è�éaênéGÜ�¸Qè�é´ê@».»b»<ûÿº¨é4'-0�}æ�¸{è�é´ê@»�óîúE¸{è�éaê@»´÷�¸{èE»<ñ ö2¸{è�éaê@»´÷�¸{èE»�ûüº'é ¸{á'$iáq»

�������������� ����������������������� ��������� "!�!�#� ���������$������%����������&('*)"+� �,�$��� ��-��.%�����������&0/1'-2234�576���#��$+�8$��'9����+�8$�:-;��</='>)"+� �,�$��� �������������� �#�#���/.?$2�3�5@2�'9���+�8$�¹

Page 3: Modeling wells in porous media flows - CORE · Modeling wells in porous media ows Pierre Fabrie, Thierry Gallouet To cite this version: Pierre Fabrie, Thierry Gallouet. Modeling wells

§s�'/,b/­Ü ð 4'/3'8q=./��=.�'/�=.-0¡�/�4'/>,.-0�:(2=b-¤�q/�8+¥ÉÜå(+3'4 §s�'/,./�4'-0��(23'4åõª4'/>3'8:=b/"=.�'/�9'�b9)(27��.�)(+=.-�(27s4¨-BA</,b/3_=.-�(278:�E/,G(2=.8q,.��$�5s�'/�¥{9'3¨¢=.-08:3¨��ò�°Eô­°'ö+°)ø2°'úE°_ý (23¨4�þ�(2,./V«:-0�:/3�(23'4�÷�-0�e(j«q-¤�q/3�¡�/n(2�b9',./m8q3 ì ¸��./>/C*E/7082§�»$5%8C¢>8:¡��¨7¤/>=./�=b�'/��',b8:*'70/¡�°:=b�'/�¥{8:707082§s-03'«�-03'-0=.-�(+7�(+3'4�*E8:9'3¨4)(2,b£�¢8q3'4'-0=.-08:3¨��(2,./m«q-¤�q/3�ç

Ü�¸Qè�éGº:»%ûåÜ�Cq¸{èE»é�è�ë"ìmé ¸{á'$i®q»ô�¸�æ}¸{è�éaê@»b».õ�Ü�¸{è�éaê@»EDFs¸{èE»%ûÿº¨én¸Qè�é´ê@»%ëHG)ìüí�¸{º¨é.ï�»>é ¸{á'$ ±_»

æ�¸Qè�é´ê@»"DFs¸{èE»%ûÿº¨én¸Qè�é´ê@»%ëHG)ìüí�¸{º¨é.ï�»�I ¸{á'$�Jq»§s�'/,b/KFs¸{èE»�-0�s=b�'/�8:9¨=a§e(+,.4�3¨8:,.¡�(27E9'3'-0=s�:/>¢=.8q,�(2=s=.�¨/x*E8:9¨3'4)(2,b£LGEìü82¥�ìÿ(+=�=b�'/��E8:-03_=eè�$M</>=s9'��«q-¤�q/�=b�'/�(2�.�b9'¡��¨=.-08:3'�}8:3�=b�'/�4)(+=@('$ïüë�¶ Û�N O}é�ì|-¤��(V*E8:9'3'4¨/4"8:�E/3��./>=�82¥%¶ ÛKPqé0Q�ûüáj8:,�®'é}§s-0=.��(RM<-0�'�b¢@�¨-¤=b¼m¢8:3_=.-03_9'8q9'�s*)8q9'3'4)(+,.£<é ¸{á'$i²q»

øsëHS0T�¸.¸{º¨é.ï�»�U*V�¸ ì}».»é'ºXW øYW ¹qéÜ C ëHS0T�¸Qìs»é)ºZWåÜ C Wä¹:é ¸{á'$�[q»

ò�ë\V�¸Þ¶ Ûjé.¶ Ûm»}-0�]M<-0�'�b¢G�'-0=.¼m¢8:3_=b-¤3_9'8q9'��(23¨4ò�¸�^:»sûåºV¥{8q,_^`Wåº'éò�¸�^:»sûä¹�¥{8q,_^`aä¹:é ¸{á'$iµq»

ýä-0�e(Rbs(+,@(+=.�'/8_4'8q,.£�¥{9'3'¢=b-08:3�¥{,.8q¡ ì|í­¸{º¨é.ï�»}í�¶ Û -03X=b8Xc P ¸Þ¶ Û�»>é��b//�¬iá:º³¯���$ ��$s¹n®q®'°q�@(+=.-0�a¥{£_-03'«Eçdfehg º'é�ýC¸Qè�é´êné@in»kjlDmjXa e_n j n ?2éw¥{8q,�('$ /:$}¸{è�éaê@»wë­ì|í­¸{º¨é.ï�»>é"opimë�¶ Û�é"oqjCë"¶ Û�P_édsr ë�¶ Û�é n ýC¸Qè�é´êné*i�» n W r éO¥Q8:,�('$ /:$}¸{è�é´ê@»�ë­ì|í­¸Qº'ébï�»é"opiÉë�¶ Û�é ¸{á'$�tq»þ�-0��(ubs(+,@(+=.�'/8_4'8q,.£�¥{9'3'¢=b-08:3�¥{,.8q¡ ì|íî¸Qº'ébï�»�í�¶ Û -03_=.8j¶ Û P �G(2=b-¤�´¥{£_-03'«)çdEvw ë"¶ Ûjé n þ�¸{è�éaêné@in» n W vw éO¥{8q,e(¨$ /:$s¸{è�éaê@»wë"ìåíO¸Qº'ébï�»éEopimë"¶ Ûjé ¸{á'$0¹�º:»

ô -0��(�¢8q3_=.-03_9'8:9'��(+3'4�*)8q9'3'4'/>4�¥Q9'3'¢=b-¤8q3�¥{,.8q¡�¶ ÛKPm=b8xc P ¸Þ¶ Û�»s(+3'4��@(2=b-0�aÄ)/�dsyzg º'é�ô�¸�{�»�jKD�jRa y|n j n ?2é"op{Éë�¶ Û P é"oqj ë�¶ Û P I ¸{á'$0¹q¹n»

}�8q=./j=.�)(+=�èxD�^�4'/3'8q=./�V=.�'/j9'�.9E(27%�.¢n(+7�(2,m�',.8_4'9¨¢=m82¥eèO(+3'4\^�-03"¶ Û�P_°�c P ¸�¶ Û�»s-0�m=.�'/j�./>=x8+¥_Q�í\Q�¡�(+=.,.-0¢/>�§s-0=.��,./�(27<¢8_/Gâ�¢-0/3_=b�x(23¨4 n D n 4'/>3'8:=b/��=b�'/�/9'¢>7¤-04'-�(+3�3'8q,.¡�-03�¶ ÛKP�(+3'4�=.�'/m-03'4'9¨¢/4"3'8q,.¡�-03zc P ¸�¶ Û�»>$¶a=�,./>¡�(2-03'�j=.8O¡�(2Àq/�¡�8:,b/��',b/¢-0�./�=.�¨/ (+�.�b9'¡��'=b-08:3'�j8:3åúq÷ (+3'4åöG÷�$~M</=��Ò¸{ìs»j4'/3¨8:=./�=.�'/��./>=�82¥mÄ)3¨-¤=b/¡�/�(2�.9¨,./�s8q3�ìm$�Ûs/>¢n(2707<=.�E(2=�(ÉÄ)3'-0=./m¡�/�(2�.9¨,./m÷­8q3�ì|-0�e(R�q�.(24'4¨-¤=b-0�:/x(+�'�'70-0¢n(2=b-08:3j¥{,b8:¡�=b�'/���8q,./7��q�´(+7¤«q/*',G(82¥�ì|-03�¶ Û�$}¶Q¥½÷ùëH�Ò¸{ìs»s=G(2Àq/��-0=.�s�:(+709'/�s-03"¶ Û O °¨=.�'/3�÷OëH� O ¸{ìs»>$s6m-0�:/3�÷�-¤3z�Ò¸Qìs»°'-0=�-¤�s-04'/>3X=b-ãÄ)/4­(+�9'�b9)(27070£m§s-¤=b�C=b�'/s70-03'/n(2,�¢8:3_=b-¤3_9'8q9'��(2�'�¨7¤-0¢n(+=.-08:3V-¤3¨4'9'¢/4j8:3RV�¸ ìs»�¸�V�¸ ìs»�-0�%/3'4'82§}/4�§s-0=.�V=.�'/]S0T�¸Qìs»��a3'8:,b¡�»>$5s�'/3�°0�Q÷�é*���@����� ��������� ��� ���mû�� � ��¸Qè)»�Qq÷�¸{èE»°E¥{8q,C(23_£\� ë�V�¸ ìs»$�}�8q=./�=.�E(2=C(�70-03'/n(2,m¢8q3_=.-03_9'8:9'�j(2�'�'70-0¢n(+=.-08:3¥{,b8:¡�V�¸ ì�»}=b8C¶ Ûä-0��3¨8:=}3'/¢/�b�@(+,.-070£�«q-0�:/3�*_£��.8q¡�/�÷"ëH�Ò¸{ìs»�¸�*¨9'=s-0=�-0�s(270§e(�£_��«q-0�:/3�*X£��b8:¡�/s÷­ëz�Ò¸ ìs»b»$5s�'/m=.8q�)8q708:«:-0¢n(+7)4¨9)(27��.�)(+¢/�82¥�(K�s(23E(2¢G�"�.�E(2¢/K� -0�s4'/3'8q=./4"*_£z���{$5s�'/�(2�b�.9'¡��'=.-08q3'��8q3"ú)°'öm(+3'4�÷­(2,b/�=.�'/�¥{8q707¤82§s-03'«

÷�ëL� O ¸{ìs» ¸{á'$0¹�á:»(23¨4

úEé@ö}ë\S�T�¸.¸{º¨é.ï�»EU*V�¸ ì�»b»úE¸{è�éaê@»�a º¨é�ö2¸{è�éaê@»�aåº'é ¸{á'$0¹�®:»

Â�¢>¢8:,b4'-03'«�=.8�=b�'/�*E8:9'3'4E(2,.£�¢>8:3'4'-0=b-¤8q3O¸{á'$�Jq»°q8:3'/m�)(+�s=.8�(2�b�.9'¡�/m=.�'/�¥{8q7¤7082§s-03'«m¢8q¡��)(+=.-0*'-070-¤=a£V¢8:3'4¨-¤=b-08:3� � úE¸{è�é´ê@»fQq÷�¸{èE»�û � � ö2¸Qè�é´ê@»fQq÷�¸{èE»I ¸{á'$0¹>±X»

á

Page 4: Modeling wells in porous media flows - CORE · Modeling wells in porous media ows Pierre Fabrie, Thierry Gallouet To cite this version: Pierre Fabrie, Thierry Gallouet. Modeling wells

¶a3�=.�'/m=b�',.//m� 4'-0¡�/3'�.-08q3)(27<¢n(+�./C(+3"(24'4'-0=b-¤8q3)(27<(2�b�.9'¡��'=.-08q3�-¤�s,b/Å_9'-0,./4�$�¶a=s,./�(24'�÷"ëO¸1� 1 � � ¸{ìs»b» � é"op� g áfI ¸{á'$0¹-J:»

¶a3�=.�'/V=a§�8�� 4'-0¡�/3'�.-08q3)(27'¢�(2�./q°)=b�'-0��(2�b�.9'¡��'=.-08:3�-0�s3'8q=�(23"(24¨4'-0=.-08:3)(+7�(+�.�b9'¡��'=b-08:3��b-03'¢/R�ä1 � � ¸Qìs»���V�¸ ìs»w¥{8q,� g á�(23'4�=.�'/3h�Ò¸{ìs»Y��¸=�ä1 � � ¸Qìs».»1�a$��å�¨/3 Q�ûÚá�(23'4�¥Q8:,�=b�'/j¡�8_4'/70-0¼n(2=b-08:3�8+¥��E8:,.8q9'�m¡�/4¨-k(V¦E82§s�n°E=.�'/¡�/�(2�.9¨,./�÷��)(+��8+¥{=./>3�-0=.�}�.9'�¨�)8q,.=},./4'9¨¢/4"=b8C�b8:¡�/s�E8:-03_=.��°:=b�'-0���b9'�'�E8:,b=�¢8q3'�.-0�b=.-03'«j-¤3�=b�'/m708X¢�(270-0¼n(2=b-¤8q3C8+¥�=.�'/§}/707¤��$m¶a3­=b�'/�=.�¨,.//@�a4'-0¡�/>3'�.-08:3E(27�¢�(2�./q°�=b�'/ (+�.�b9'¡��'=b-08:3�¸Qá'$0¹�Jq»s-0��,b/n(270-0�.=b-¤¢V¥{8q,�=.�¨/�(2�'�'70-0¢n(+=.-08:3'��°)(+=x70/n(+�.=�¥{8q,¦)82§s�s=b�',.8q9'«:�"�E8:,b8:9'��¡�/4¨-k(¨$%¶a3'4'//>4�°<(j¢8:3_�:/>3'-0/3_=�¡�8_4'/>7¤-0¼n(+=.-08:3�8+¥�=.�'/�§}/7070�s70/n(24'��=.8�(j¡�/n(2�b9',./m÷O§s�'-0¢G�¢8q3'�.-0�.=b��-03�(�Ä)3'-0=./s3_9'¡�*E/,%82¥<=./,b¡��%�.9¨¢@��(+�¢¡�£j§s�'/,./]¡�-0�½(�*E8:9'3¨4'/4�¡�/n(2�b9',@(+*'70/%¥{9'3¨¢=.-08:3�(+3'4R£j-¤�%=b�'/s8:3'/4'-0¡�/>3'�.-08:3E(27fM</*E/�b«:9'/�¡�/n(2�b9',./s8q3�(m8q3'/�4¨-¤¡�/3'�b-¤8q3'3)(+7'¢9',b�:/�¸�-03'4'/>/4�°_-0=�§s-0707'*)/m�b9_â�¢-0/3_=�¥Q8:,�9'�s=b8C(2�b�.9'¡�/=.�E(2=¢¡ ë¤S�?q¸{ìmé@£E».»$ à 9'¢G��(m¡�/�(2�.9¨,./s*E/708:3'«q��=.8�¸=�ä1 � � ¸{ìs»b»1�¨¥{8:,0� g á��b-¤3¨¢/�=b�'/,./m/Á_-0�.=b��(m¢8q3_=.-03_9'8:9'�V¾.=.,G(2¢/8:�E/,G(2=.8q,.¿j¥{,b8:¡¥�ä1 � � ¸Qìs»}=.8xS � ¸�¦%»°E§s�'/3H¦�-0��(��b/«:¡�/3_=s-03 ìm$§R¨s©«ªf¬�­¯®E°�±³²Z´�µ�¶\µ�·f¸(µRµ¹·q¶\º@´�»½¼f´�¾¿�µ�¿�´�ÀÁ´$ÂHµ�·f¶«¿ÃÀĶ@º�µ�¶@ÅxÆ�Çs¿�Å~¿�¾xÈ�¿�É-¶-ÀsÊx¿�µK¸m¼�¼f¶*¸(Ë@¾\¿�À̵¹·q¶«µ�¶-Ë» øúq÷Í¿�ÀÎ�Ï-Çs¸�µ�¿�´�À¯Ð=Ñ�Ò�Ó�Ô�Ê]Õ�Çsµ�µ�·f¶�º*´(»�¼f´�¾¿�µ�¿�´(À\´�ÂRµ¹·q¶Y¼pË$´-Å�Çsº@¶@ÅKÆ�Ç�¿ÃÅH¿�¾u¸(À«Ç�À(Ö�Àf´(×0ÀsÊ�¿�µ�¸@¼(¼Ø¶@¸�Ëm¾u¿�À«µ�·f¶Rµ�¶�Ë»�ÜEöG÷Á¿�ÀÎ�Ï-Çs¸�µ�¿�´�À¯Ð=Ñ�Ò�Ó�Ô�Ò«Ù�·f¿�¾K¾µ�ËÇsº�µ�ÇsË$¶R´�Â<Î�Ï-Çs¸�µ�¿�´�À~Ð=Ñ�Ò�Ó*Ô>Ê]µ�´@È�¶-µ¹·q¶�ËÚ×0¿�µ�·hµ¹·q¶Ú¾¶*º@´(ÀfÅÚ¼f¸�˵�´�ÂKÎ�Ï�Çs¸(µ�¿�´(À¯Ð=Ñ�Ò�Ñ�ÔÛÐ�µ�·f¸�µ¿�¾�4¨-¤�_æ û úq÷�ó ö ÷�ÔL¸�ÀfÅÝÜ0´�ÀfÅ(¿�µ�¿Ã´�À¯Ð=Ñ�Ò�Þ�Ô¤´(À"ò³Ð�¸�ÀqÅ�ºzW Ü C W ¹mÔ>Ê]¿�¾uº�Ë�Ç�º-¿�¸�ß7µ�´H´(Õ�µ�¸�¿�À«µ�·f¶R¶@à(¿�¾µ�¶-Àfº@¶x´�ÂR¸¾´�ß�Ç�µ�¿�´(Àz¾Ç�º@·Hµ�·f¸(µ�ºXW ÜzWÿ¹(Ò&¨8:,C(�¡�/n(2�b9',./z£�8:3�=.�'/���8q,./7]��ó�(270«q/*',@( 8:3 ìm°%8q3'/�4'/3¨8:=./>�xS�á<¸{ìmé@£)»j¸ãßåë ¬iº'é@âù¯{»s=.�¨/�9'�b9)(27]S�á��b�)(2¢/,./>7k(+=./4 =b8 £�$�¶Q¥<£�-0�j=.�'/�9'�.9E(27]M</>*)/�b«:9'/�¡�/n(+�.9',b/­¸�4¨/3'8:=b/4 *_£¯ã -03O=.�'/��./>ÅX9¨/7�»°}=.�'/��.�)(+¢/HS�á<¸{ìmé@£E»m-0�4'/3¨8:=./>4"*_£xS�á<¸{ìs»>$ô�¨(ä7F�å�æ�å�çfF¯®E°�±èß¹¶-µ¢Ð=Ñ�ÒÃé@Ô�ê*Ð=Ñ�Ò�Ó�ë@Ôu·f´�ß¹Å�ÒRÙ�·q¶�À"¸QÜ�éQß)»Z¿�¾<¸�×0¶@¸�Ö`¾´�ß¹Çsµ�¿�´�ÀÛ´$ÂuÐ=Ñ�Ò�Ó�Ô�ê*Ð=Ñ�Ò�ëmÔ`¿ Â

Ü�ëHS�?q¸.¸Qº'ébï�»U@ì�1¸{ìs».»>é�ºZWåÜ�W ¹qéÜ�ëHS0T�¸b¸{º'ébï�»U@S}1n¸Qìmé.÷�»b»é�ºZWåÜ�¸Qè�é´ê@»�W ¹qé\´(Ës÷�ê$¸�Òö�Ò}è�ë"ìmé«Â´�Ë�¸�Òö�Òwê�ëO¸{º¨é.ï�»>é ¸{á'$0¹�²:»

Ü ð ë¤S ? ¸.¸Qº'ébï�»U�¸1� 1 � í ¸{ìs»b» � »é"opi g QEé ¸{á'$0¹-[:»Ü�ë¤V�¸b¬iº'ébï}¯=U�¸1� 1 � í ¸Qìs».» � »>é"opi g QEé�Ü�¸�I¤éGº:»�ûÿÜ�Cm¸$¿ÃÀz� 1 � í ¸Qìs».» � »>é ¸{á'$0¹�µ:»

ß�ëHS T ¸.¸Qº'ébï�»U�� 1 � � ¸Qìs».»>é�op�Éëî¬0¹:éGá:»>é ¸{á'$0¹-t:»

� Ü ð ¸$I0éaê@»>é*��� �¹î�ï�ð�ñ1� � � î�ï�ð�ñ ñ � � ô�¸Þæ�¸{è�é´ê@».».õ�Ü�¸{è�é´ê@»�DnõR�}¸{èE»$Q_èó � � æ�¸{è�é´ê@»�DnõR�}¸{èE».ò�¸{Ü�¸{è�éaê@»b»$Q_èmñ � � Ü�¸Qè�é´ê@»$�}¸{èE».ö+¸{è�éaê@»�Q:÷�¸Qè)»·ó � � ø2¸{è�é´ê@»$�}¸{èE».úE¸{è�é´ê@»$Qq÷�¸{èE»�ûüº'é

op�ùëh�ä1 � ò ¸{ìs»é"o�ó g QEé�´�ËK¸(Ò�¶-Ò�ê}ëî¸Qº'ébï�»é¸{á'$iáqº:»

æ}¸{è�éaê@»�ûäósýj¸{è�éaênéGÜ�¸Qè�é´ê@».»·¸�õmß�¸{è�é´ê@»<ó�þ�¸{è�éaênéGÜ�¸Qè�é´ê@».»b»'é«Â´(Ë�¸�Ò�¶-Òm¸{è�éaê@»wë"ìüí­¸{º¨é.ï�»>éó � � æ}¸{èE»�DõRôs¸{èE»$Q_è�û � � ôs¸Qè)»búE¸{è�éaê@»�Q:÷�¸Qè)»�ó � � ôs¸Qè)»bö2¸{è�é´ê@»$Qq÷�¸{èE»é

o�ô�ë\�ä1 � ò ¸Qìs»é"opó g QEé0´(Ë�¸�Ò�¶-ÒwêeëO¸{º¨é.ï�»�I¸{á'$iᨹn»

}�8q=./m=.�E(2=sæOëHS0T�¸.¸Qº'ébï�»U�¸�S � ¸{ìs».» P »w¥{8:,s(2707.�VëO¬¤¹qé@áq»$5s�'/m¡�(+-03�,./�b9'70=�82¥%=.�'-0�s�)(+�)/>,�-0��=b�'/m¥{8q707¤82§s-03'«m8q3'/:ç

®

Page 5: Modeling wells in porous media flows - CORE · Modeling wells in porous media ows Pierre Fabrie, Thierry Gallouet To cite this version: Pierre Fabrie, Thierry Gallouet. Modeling wells

õ�ö�¨sç�¬�¨s©÷®E°�±ùø.¶-µ_Ð=ÑsÒ�é@Ô�ê*Ð=Ñ�Ò�Ó�ë@Ôu·f´�ß¹Å(ÒXÙ�·q¶�Ë�¶u¶@à(¿�¾µ�¾�¸�µ�ß¹¶@¸�¾µ�´�Àf¶Ú×0¶@¸�Ö`¾´�ß¹Çsµ�¿�´�ÀÛµ�´zÐ1ÑsÒ�Ó�Ô�ê*Ð=Ñ�ÒÃë@Ô�Ê0¿�ÀHµ�·f¶Z¾m¶-Às¾¶´$ÂYúK¶�û0Àf¿�µ�¿�´�ÀÛÑ�Ò�Ó�Ò§R¨s©«ªf¬�­¯®E°¹®ÍÜ0´(ÀfÅ�¿�µ�¿�´(À Ð=Ñ�Ò�Ó�ë@Ôx¸�ß�ß¹´(×�¾KÇ�¾Úµ�´�´�Õ-µ�¸(¿�À\¸`¾µ�Ë�´�À�È(¶�ËZ¶*à(¿�¾µ�¶-Àfº@¶ÚË$¶�¾mÇ(ß�µ_Ð�¿ Â]Qjûü®-Ô]´�ËÚÐ1ÑsÒ�Ó�Ô�ê*Ð=Ñ�ÒÃë@Ô�ÒÙ�´uÕ@¶K»Ú´�Ë�¶�¼pË$¶@º-¿�¾¶Ê�µ¹·q¶�¼pË$´-´$ÂY´$ÂRÙ�·f¶@´(Ë$¶-»üÑsÒ�ÓYÈ�¿�É-¶�¾*Ê0ÇsÀfÅ(¶�ËYµ¹·q¶]·fý�¼f´�µ�·f¶�¾¶¾KÐ1ÑsÒ�é@Ô�ê*Ð=Ñ�Ò�Ó�ë@Ô�Ê0¸�Àz¶@à�¿�¾mµ�¶-Àfº@¶KË$¶�¾Ç�ß�µ×0¿�µ�·�Ü ð ë\S ? ¸.¸Qº'ébï�»U�¸1� 1 � í ¸{ìs».» � »K¸�ÀqÅ Ü ë\V�¸.¬iº¨é.ï}¯$U�¸1� 1 � í ¸Qìs».» � »�´(Ë�¸�ß�ß�i g ásÊ�¿�Às¾µ�¶@¸(Åx´$Â�´�ËK¸�ß�ß�i g Q�ÒYþ�¶@¶µ�·f¶½¼pË$´-´$Â�´�ÂxÙ�·f¶@´�Ë�¶�»ÿÑ�Ò�ÓR¿�Àzþ�¶*º-µ�¿�´�ÀHë(Ò¶a3�8:,.4¨/,�=.8��',b82�:/j5s�'/8q,./¡ªá'$0¹�¸�-03 à /¢>=.-08:3¯Jq»°<8q3'/�ÄE,.�.=V�',.82�q/�m=.�'/�/Á_-0�.=b/3'¢/�82¥e(��b8:709'=.-08q3"=.8O¸{á¨$0¹n»��¸{á¨$�J:»§s�'/3�=.�'/s4E(2=@(V(2,./s¡�8:,b/�,b/«:9'7�(+,�¸�-03 à /¢>=.-08:3�±X»°+3)(2¡�/70£�§s�'/3�÷�ëHS�?:¸Qìs»$%5s�'-0�%/Á_-0�.=b/3'¢/m,./>�.9'70=n°q¥Q8:,�,b/«:9'7�(+,4)(+=@('°_,b/n(24¨�x(2�}¥Q8:707082§s�nç��¬�ç��0ç�{å�æ�å�ç�F ®E°�±³ø"¶�µ]Ð=Ñ�ÒÃé@Ô�ê*Ð=Ñ�Ò�Ó��-Ôx·f´�ß�ÅH¸�ÀqÅH¸�¾@¾Çs»u¶V÷�ë¯S�?q¸{ìs»�Ò Ù�·q¶�Ë�¶R¶*à(¿�¾µÃ¾u¸z¾´�ß�Ç�µ�¿�´(ÀO¸{Ü�é{ßE»K´$ÂxÐ=ÑsÒ�Ó�Ô�êÐ=Ñ�Ò�ëmÔR¾m¸(µ�¿�¾�Â�ý�¿�À�È��

Ü�ëHS ? ¸b¸{º¨é.ï�»�U*ì 1 ¸Qìs».»��zV�¸b¬iº'ébï}¯=U@S ? ¸{ìs».»>é�ºZWåÜ�W ¹qéÜ ð ë\S�?2¸.¸Qº'ébï�»U�¸�ì�1�¸{ìs».»=��»é ¸{á'$iáqá:»

Ü�¸$I0é@ºq»�ûüÜ C ¿ÃÀxS ? ¸{ìs»é ¸{á'$iáq®:»

ß ë¤S T ¸b¸{º¨é.ï�»�U*ì 1 ¸{ìs».»>é%æOëHS T ¸.¸Qº'ébï�»U�¸�S ? ¸Qìs».»�P2»é ¸{á'$iá+±X»

� ܨð@¸�I¤é´ê@»é@��� �Yïk� � � Yï ñ � � ô­¸Þæ�¸Qè�é´ê@».»bõ�Ü�¸Qè�é´ê@»�DnõR�}¸Qè)»�QXèó � � æ�¸{è�é´ê@».D�õR��¸Qè)»bò�¸QÜ�¸Qè�é´ê@».»�Q_èxñ � � Ü�¸Qè�é´ê@»$�}¸{èE».ö+¸{è�éaê@»´÷�¸{èE»$Q_èmó � � ø2¸Qè�é´ê@»$�}¸{èE».úE¸Qè�é´ê@»a÷�¸Qè)»�QXèjûüº'éop�ùëHì�1³¸Qìs»é0Â�´�ËK¸(Ò�¶�Ò½êsëî¸Qº'ébï�»é

¸{á'$iá(J:»æ�¸Qè�é´ê@»%ûüósýC¸Qè�é´êné@Ü�¸{è�é´ê@».»��nõmß�¸{è�éaê@»·ó­þ�¸Qè�é´êné@Ü�¸{è�é´ê@».» �%é ¸{á'$iáq²:»

ó � � æ}¸{è�éaê@»"DnõRôs¸Qè)»�Q_è�û � � ôs¸{èE».úE¸{è�é´ê@»a÷�¸{èE»$Q_èVó � � ôs¸{èE».ö+¸{è�éaê@»´÷�¸{èE»$Q_è�éopô�ëHì�12¸{ìs»>é0´�Ë�¸(Ò�¶�ÒwêsëO¸{º'ébï�»I ¸{á'$iá([:»5s�'/}�',.8_8+¥)82¥)#%,b8:�E8:�.-0=b-¤8q3Cá'$0¹%-0��¢�(2,.,b-0/4C8q9'=�§s-0=b��=.�'/ à ¢G�)(+9'4'/,�© ��Ä)Á_/4j�E8:-03_=�=.�¨/8:,b/¡­$�à 8:,./>82�:/,%§}/�/�.=G(2*'70-0�.��.8q¡�/s/�b=.-0¡�(+=./�w8:3�=b�'/��b8:709'=b-¤8q3'�}«:-0�:/3�*_£�#%,.8q�)8q�.-0=.-08q3�á'$0¹s£_-0/704'-03'«m=.�¨/��',b8X8+¥�8+¥�5s�'/8:,b/¡ á¨$¤¹�9'�)8q3��)(+�.�b-¤3¨«=.8�=b�'/m70-¤¡�-0=�8q3"(2�'�',b82Á_-¤¡�(2=b/s�.8:709'=b-08:3'��$

Í�Î´Í � Ö�Ö�Õ����������bÑ����¶a3�=.�'-0�s�b/¢=.-08q3�§}/m«:-0�:/m=a§}8 (+�'�'70-0¢n(2=b-08:3'�s(2,b-¤�b-03'«�¥Q,.8:¡ �)/=b,.8q7¤/>9'¡ />3'«:-03'//>,.-03'«"¬i®�¯.°<¬ ±�¯.°<¬ ²>¯.°<¬iµ>¯.$%5s�'/�Ä),.�b=s8:3'/4'/n(+70�C§s-0=.�O¡�-0�.¢-0*'70/V¦)82§s��=.�¨,.8:9¨«:���E8:,b8:9'�j¡�/>4'-�('°<=b�'/��b/¢8:3¨4 8q3'/�4'/n(+70�C§s-0=.�O-0¡�¡�-T�b¢-0*'70/m¦)82§s��$"¶a3�*E8:=b�(2�¨�'70-¤¢�(2=.-08q3'��=b�'/m¡�/n(+�.9',b/�s=./>,.¡��s¡�8_4'/70-0¼/�=b�'/m§�/7070�n$

®E°¹®�°Ã± �Áå1{���å��! �¨#"0ç�$Â�¢>¢8:,b4'-03'«�=.8�¬ [>¯.°2¡�(2�b��=b,@(+3'�a¥{/>,�8+¥·(m�b9'*'�b=.,@(+=./�§s-¤=b��¢8:3¨¢/3_=.,G(2=.-08q3�ø}-¤3�(m�G(2=.9¨,@(2=b/4��E8:,.8q9'�}¡�/4'-09'¡ä�G(2=b-¤�´Ä)/�=.�¨/�¥Q8:707082§s-¤3¨«�/Å_9)(2=b-08:3'��¸Þ¡�(2�b��¢>8:3'�b/,.�:(+=.-08:3"(23¨4&%m(2,b¢£�7�(�§�»$

øGðG¸{è�éaê@»<ñù4'-0��¸Þæ�ø»>¸{è�éaê@»b»�óO4'-0�!'�'1ã�¸Qø»�ñ�ô­¸Þæ�»)(³õ�ø (_¸{è�é´ê@»*(ñxø+¸{è�éaê@»'úE¸Qè�é´ê@»a÷�¸Qè)»·ó vø2¸Qè�é´ê@»'ö2¸Qè�é´ê@»a÷�¸Qè)»%û庨és¸{è�éaê@»wë"ìüí­¸Qº'ébï�»é ¸{á'$iáqµ:»

æ�¸{è�é´ê@»)ñ + ¸Qè)»£�¸{ø2¸Qè�é´ê@».» ¸{õmß�¸{è�éaê@»�ó-,)¸{ø+¸{è�éaê@»b»aþ�»�û庨és¸{è�éaê@»wë"ìüí­¸Qº'ébï�»é4'-0�Xæ}¸{è�éaê@»·ó�úE¸{è�éaê@»´÷�¸{èE»)ñ ö2¸Qè�é´ê@»a÷�¸Qè)»%ûüº'é}¸{è�éaê@»wë"ìüí­¸{º¨é.ï�»>é

¸{á'$iá(t:»

±

Page 6: Modeling wells in porous media flows - CORE · Modeling wells in porous media ows Pierre Fabrie, Thierry Gallouet To cite this version: Pierre Fabrie, Thierry Gallouet. Modeling wells

§s-0=.��=.�¨/�¥Q8:707082§s-¤3¨«x-03'-0=b-k(+7�(+3'4�*)8q9'3'4)(+,.£�¢8:3'4¨-¤=b-08:3'�

ø2¸Qè�éGº:»%ûåø C ¸Qè)»>é�è�ë ìmé ¸{á'$i®qº:»¸�ã�¸Qø2¸{è�é´ê@».»<ñ ô�¸�æ�¸Qè�é´ê@».»b»_õ�ø2¸{è�éaê@»"DF�ûÿº¨és¸{è�éaê@»wëHG)ì|íO¸{º'ébï�»é ¸{á'$i®¨¹n»

æ�¸Qè�é´ê@».D�F­ûüº'és¸{è�é´ê@»�ëHGEìåíO¸{º¨é.ï�»�I ¸{á'$i®qá:»

¶a3 =.�'-0��¡�8_4'/7a°%=.�¨/"¢8:3¨¢/3_=.,G(2=.-08q3å82¥�=.�¨/"�.9'*¨�.=.,G(2=�-¤��4'/3'8q=./4|*X£ ø�(23'4 =.�¨/"«:-0�:/>3 ¢>8:3'¢/>3X=b,@(+=.-08:3ÿ(+=�=.�'/-03/.G/¢>=.-08:3�§}/707'-0��4'/3¨8:=./>4�*X£ vø2$�5s�'/sÄE7¤=b,@(+=.-08:3j�:/708_¢-0=a£�-0��4'/>3'8:=b/4�*_£�æ­(+3'4�=.�¨/s�',./�b�.9',b/�*_£�ß�$��O/}3'8q=./]ã�¸{ø»=.�¨/x¡�8:70/¢9¨7k(+,%4'- A�9'�b-08:3�°'ô�¸�æ%»�=b�'/�4¨-¤�b�)/>,.�.-08q3�=b/3'�b8:,n° + =b�'/��E/,.¡�/n(2*¨-¤70-0=a£j=./3'�b8:,�°�£�¸{ø»%-0�}=.�'/m�_-0�.¢8q�.-0=a£_°0,E¸{ø»4'/3¨8:=./>��=.�¨/x4¨/3'�.-0=a£�(23'4�þ"-0�s=.�'/m«:,G(��_-¤=a£��q/¢=b8:,n$�}�/Á_=�úq÷î(+3'4�öG÷�,./>�',./�b/3_=m=.�'/�¦)82§å,G(2=b/��(2=s=b�'/m-¤3�.G/¢=b-¤8q3(23¨4��¨,.8_4'9'¢=b-08:3�§�/>7¤70��°'ú:÷O(23¨4�öG÷��@(+=.-0�a¥{£�¸{á¨$¤¹�á:»k� ¸Qá'$0¹n®q»�(23'4�=.�'/V¢8:¡��)(2=b-¤*¨-¤70-0=a£m¢8:3¨4'-0=.-08:3O¸{á¨$0¹±X»w-0��¥Q9'7ãÄ)707¤/>4�$5s�'/�¥{8q7¤7082§s-03'«��_£_�E8:=.�¨/�.-0��(2,b/m�'7�(2¢/4�8:3�=b�'/�4)(+=@(

ã�ëHV�¸�¶ Û�é.¶ Ûx»>é d ã21 g ºfUfã21èWùã�¸��<»éso��"ë�¶ Û�éô�û ¸�Q43652»*3 � 5/7 1 �686868 � P éfQ4365më\V:9n¸Þ¶ ÛKPqé.¶ Ûx»�U)ô�¸�æ%»�jKD@jÚa º¨é(o)æOë�¶ ÛKPqé(ofj�ë�¶ ÛZP:é ¸{á'$i®q®:»

§s�'/,b/uV:9n¸���é ;�»�4¨/3'8:=b/�m=.�'/m�b/=�8+¥�¢8q3X=b-03X9¨8:9'��(+3'4�*)8q9'3'4'/4�¥{9¨3'¢=.-08q3'�}¥{,.8q¡�� =.8<;�$+ û ¸�= 3 5 » 3 � 5�7 1 �686868 � P é�= 3 5 ë\S0T�¸{ìs»�U dfehg ºfU + ¸Qè)»kjuD�jXa e é(ofj�ë�¶ Û P é�(fI /�I_-03"è�ë"ìmé£�ëHV�¸�¶ Û�é.¶ Ûm»>é dsrhg ºfU r W�£�¸��<»>é(op��ë�¶ Û�é,�ëHV�¸Þ¶ Ûjé.¶ Ûm»>é ¸{á'$i®+±X»

vøsëHS0T�¸.¸{º¨é.ï�»�U*V�¸ ì}».»é'ºXW vøYW ¹qéøC�ëHS0T�¸{ìs»>é)ºZW ø�CYW ¹(I ¸{á'$i®(J:»

ô�¨(ä7F�å�æ�å�çfF¯®E°¹®èø"¶�µ0Ð=Ñ�Ò�émÔ�Ê�Ð=ÑsÒ?>0>�Ô>ê*Ð=ÑsÒ?>�ëmÔX¸�ÀqÅxÐ1ÑsÒ�Ó-Ñ�Ô�ê*Ð1ÑsÒ�Ó�ëmÔu·f´�ß�Å�ÒÚÙ�·f¶-À�¸Qø2éQß)»Y¿�¾<¸R×0¶@¸-ÖK¾´�ß¹Çsµ�¿Ã´�Àz´$ÂKÐ=Ñ�Ò�Ñ(ÞÔ�êÐ=Ñ�Ò?>(ÑÔx¿ Âø�ëHS ? ¸.¸{º¨é.ï�»�U*ì 1 ¸Qìs».»>é)ºZW øYWü¹:é

ø�ë\S T ¸.¸Qº'ébï�»U@S 1 ¸Qìmé.÷�»b»é'ºÚW ø+¸{è�éaê@»�Wä¹:é0´(Ë}÷�ê$¸�Ò�¶-Òsè�ë­ìmé0´�Ë�¸(Ò�¶�ÒwêsëO¸{º'ébï�»éø ð ëHS ? ¸b¸{º¨é.ï�»�Un¸=� 1 � í ¸{ìs»b» � é(opi g QEé

ø�ëHV�¸.¬iº'ébï}¯$Un¸=� 1 � í ¸{ìs».» � »é(opi g QEé�ø2¸�D¤éGº:»�ûÿø C éß�ëHS T ¸.¸{º¨é.ï�»�U� 1 � � ¸{ìs»b»é"op�VëO¬0¹:éGá:»é

¸�ÀqÅ�aø ð ¸$I0é´ê@»é@��� �¹î�ï�ð @1� � � î�ï�ð @ ñ � � ¸�ã�¸Qø2¸{è�é´ê@».»<ñ ô�¸�æ�¸Qè�é´ê@».»b»_õ�ø2¸{è�éaê@»"DnõR�}¸Qè)»�QXèó � � æ}¸{è�éaê@»"DnõR�}¸Qè)»bø2¸{è�éaê@»�Q_èxñ � � ø2¸Qè�é´ê@»$�}¸Qè)»bö2¸{è�éaê@»�Qq÷�¸{èE»ó � � vø+¸{è�éaê@»���¸Qè)»búE¸{è�éaê@»�Q:÷�¸Qè)»�ûÿº¨éop�ùë\�ä1 � � ¸Qìs»éso�� g Q)é�´�Ë�¸�Òö�Òwêsëî¸Qº'ébï�»é

¸{á'$i®q²:»

æ�¸{è�é´ê@»�û ó + ¸{èE»£�¸Qø2¸{è�é´ê@».» ¸{õmß�¸{è�éaê@»·óA,)¸{ø+¸{è�éaê@»b»aþ�»Eé¢Â�´�Ë�¸�Ò�¶-Òm¸{è�éaê@»wë"ìåíO¸Qº'ébï�»éó � � æ�¸{è�é´ê@»�DõRôs¸{èE»$Q_è�û � � ôs¸Qè)»bú)¸Qè�é´ê@»a÷�¸Qè)»�Q_è�ó � � ôs¸{èE».ö2¸Qè�é´ê@»a÷�¸Qè)»�Q_è�éopô�ëh�ä1 � � ¸{ìs»>é(op� g QEé0Â�´�Ë�¸�Ò�¶-Òwê�ë�¸Qº'ébï�»I

¸{á'$i®([:»

J

Page 7: Modeling wells in porous media flows - CORE · Modeling wells in porous media ows Pierre Fabrie, Thierry Gallouet To cite this version: Pierre Fabrie, Thierry Gallouet. Modeling wells

5s�'-0���b£_�.=./>¡ -0���q/,.£å�b-0¡�-07�(2,�=b8�=.�E(2=�82¥�=.�¨/­¡�8X4¨/7��',b8:*'70/¡�$ü5s�'/­8q3'70£ 4'- A�/,b/3'¢/O70-0/��-¤3 =b�'/­4¨-BA<9'�b-¤8q3¢8_/Gâ�¢-0/3_=m-03CB%Å_9)(2=b-¤8q3�¸{á¨$iá:µ:»}§s�¨-¤¢G�"¢>8:,.,b/�.�E8:3¨4'�m=.8&B½ÅX9E(2=.-08q3�¸{á¨$¤¹�»}-03�=b�'/j¡�8_4'/7<�',b8:*'70/¡�°'3'8q=./�=b�)(2=�=.�'/9'3'À_3'82§s3V4'/3'8q=./4j�'/,b/s*X£�ø%¢>8:,.,b/�.�E8:3¨4'��=.8mÜ�-03�=b�'/}¡�8_4'/7q�',.8q*'70/¡­$E5s�'/}4'- A�9¨�.-08:3�¢>8X/ â�¢-0/3_=�-0��ã�¸Qø»ñjô­¸Þæ�»-03'�.=b/n(24�82¥�ô�¸�æ%»}¥{8q,�=b�'/C¡�8_4'/7��¨,.8:*¨7¤/>¡­$s5s�'-0��4'/�E/3'4'/3¨¢/�8q3­øm82¥}=.�'/j4'- A�9¨�.-08:3"¢>8X/ â�¢-0/3_=x4¨8X/>�x3¨8:=�70/n(+4=.8�(+4'4'-0=.-08q3'3)(27E4'-ãâ�¢9'70=b-¤/>�e(+3'4�§�/m8:*¨=@(2-03�=.�'/m¥Q8:707082§s-¤3¨«x=b�'/8q,./¡�§s�'-0¢G��-0�s�',b82�:/4�-03 à /¢=b-08:3HJ'çõ�ö�¨sç�¬�¨s©÷®E°¹®ED�¾@¾Ç�»Ú¿�À�È�Ð1ÑsÒ�é@Ô�Ê0Ð1ÑsÒ?>0>�Ô�ê*Ð1ÑsÒ?>�ëmÔX¸(ÀfÅxÐ=Ñ�Ò�Ó>ÑÔ�ê*Ð=Ñ�Ò�Ó�ë@ÔXµ�´K·f´�ß¹Å�Ê|µ�·f¶-Ë$¶�¶*à(¿�¾µÃ¾]¸�µ�ß�¶*¸�¾µ|´(Àf¶�¾´�ß�Çsµ�¿�´(À´$ÂÚÐ1ÑsÒ�Ñ�Þ�Ô�ê*Ð1ÑsÒ?>�Ñ�Ôx¿�ÀHµ�·f¶Z¾m¶-Às¾¶Ú´$Â]úZ¶�û0Àq¿�µ�¿Ã´�ÀÛÑ�Ò�ÑsÒ®E°¹®�°�® Fm©«©«åk{���å��G �¨#"0ç�$5s�'/m¡�8_4'/>7E¥{8:,}-03'¢8:¡��',./>�.�.-0*'70/m-0¡�¡�-0�.¢>-¤*¨7¤/w¦)82§ü-03�(j�'8q¡�8:«q/3'/8q9'���E8:,b8:9'��¡�/4'-09'¡Ú,b/n(24'�

ið¸Qè�é´ê@»<ñO4¨-¤�<¸Þæ�Hwò�¸�i�».»>¸{è�éaê@»½óO4'-0��¸�¡�¸�in»bõRin»>¸{è�éaê@»ós4'-0��¸Ã­�¸�i�».»>¸{è�éaê@»�ñ iq¸{è�é´ê@»'ö2¸{è�é´ê@»a÷�¸{èE»·óîúE¸{è�éaê@»´÷�¸{èE»%ûÿº¨és¸{è�éaê@»wë"ìåíO¸{º¨é.ï�»>é ¸{á'$i®qµ:»

æ H ¸Qè�é´ê@»)ñ~cÚ¸�i:¸Qè�é´ê@».»>¸{õJI%¸{è�é´ê@»�ó ,)¸�i:¸Qè�é´ê@».»´þ�» û º'é}¸{è�éaê@»wë­ì|í­¸{º¨é.ï�»>é4¨-¤�<¸Þæ H »>¸{è�éaê@»·ó�úE¸{è�éaê@»´÷�¸{èE»<ñ�ö2¸Qè�é´ê@»a÷�¸Qè)»�û º'é}¸{è�éaê@»wë­ì|í­¸{º¨é.ï�»>é ¸{á'$i®(t:»

§s-0=.��-03'-0=.-�(27E(23'4�*E8:9'3'4E(2,.£�¢8:3¨4'-0=.-08:3'�

i:¸Qè�éGº:»%ûÁi�Cq¸{èE»é'è�ë­ìmé ¸{á'$ ±_º:»¸�¡�¸�iq¸{è�éaê@»b».õRiq¸{è�éaê@»<ñ ­�¸�iq¸{è�é´ê@».».»�D@F­ûüº'é<¸{è�é´ê@»�ëHGEìüíî¸Qº'ébï�»é ¸{á'$ ±E¹n»

æ H ¸Qè�é´ê@».D�F­ûüº'é<¸{è�é´ê@»�ëHGEìüí­¸{º¨é.ï�»�I ¸{á'$ ±_á:»¶a3�=.�'-0�m¡�8_4'/7a°Eæ H -0�m=.�'/j=.8q=@(+7��:/708_¢-0=a£_°KIù-0�x=b�'/�=.8q=@(+7��',./>�.�.9¨,./:°0i�-¤�m=b�'/C�G(2=b9',@(+=.-08:3�82¥s§e(+=./,�°�ò�¸�i�»�-0�m=.�'/¥{,G(2¢=b-¤8q3)(27¨¦)82§å8+¥�§s(2=b/,n°f¡�¸�i�»}=@(+À:/�s-03_=.8C(+¢¢8q9'3_=s=.�'/m¢n(+�'-07¤7�(+,.£j/@A�/>¢=.��°�­�¸�in»}-0�s(�«q,@(��_-0=a£C=b/,.¡�° ,C-0��(V¡�/n(+3'�4'/3¨�.-0=a£X°'(+3'4�þ"-0�s=.�'/V«:,@(��_-0=a£�¬ ²³¯.°<¬ µ>¯.$Â���¥{8q,%=.�'/}¡�-0�.¢>-¤*¨7¤/�¦)82§s��°2ïüë�¶ Û�N O}°:ìO-0��(s*E8:9'3¨4'/4�8:�E/3j�./=w82¥)¶ ÛZPm¸�Q�ûüás8q,½®q»�§s-¤=b�C(�M<-0�'�b¢@�¨-¤=b¼�¢>8:3_=.-03_9'8:9¨�*E8:9'3'4E(2,.£_°<(23'4Oúq÷ (+3'4îö ÷O,./�',b/�./>3X=j=.�¨/j¦)82§ä,@(+=./�C(+=m=.�'/j-03/.G/¢=b-08:3O(23'4"�¨,.8_4'9'¢=b-08:3O§}/707¤��°E=.�'/£��)(��:/j=.�'/¥{8q,.¡ ¸Qá'$0¹náq»��¸{á'$0¹-J:»$·5s�'/m¢8:¡��)(2=b-¤*¨-¤70-0=a£�¢8q3'4'-0=.-08q3­¸{á¨$0¹±X»w-0�s4'9'/m=.8�=b�'/�*E8:9'3¨4)(2,b£�¢8q3'4'-0=.-08:3�¸{á¨$ ±_á:»>$§R¨s©«ªf¬�­¯®E°ML¥Ù�·q¶<¸(Å�Å(¿�µ�¿Ã´�Àf¸�ßpÅ(¿ N~º�Ç(ß¹µ�ýK×0¿Ãµ¹·RË�¶¾�¼f¶@º�µ"µ�´Úµ¹·q¶�»u´-Å�¶�ß�¼�Ë$´(Õß�¶�» ß¹¿�¶�¾]¿�Àxµ�·f¶�¸(º�µ"µ¹·q¸�µ�µ�·f¶�¼Ø¸(Ë$¸(Õ*´�ß¹¿�º¶@Ï�Çs¸(µ�¿�´(À«¿�¾uÅ(¶kÈ(¶-Àf¶�Ë�¸�µ�¶R¾¿�Àfº@¶zÐ$¿ÃÀ\µ�·f¶EO�Ë$¶@¸�ß7»Ú´-Å�¶�ßQP�ÔHµ�·f¶]Â�Ç�Àqº�µ�¿�´�ÀH¡�¸�i�»RÉ-¸�Àf¿�¾=·q¶¾�´(Ë<ijû ¹L¸(ÀfÅziCû ºfÒJR.´(˾´�»Ú»u¶Ú¶@à�¿�¾µ�¶�Àqº*¶Ú¸�ÀqÅxß�´�º@¸�ß.Ë$¶1È�Ç(ß�¸�Ë¿�µ�ýÚË$¶�¾mÇ(ß�µ�¾@Ê|×�¶uº@¸�À¤¾m¶@¶TS$Ó*UmÊ!S�Ñ�UK¸(ÀfÅJS�ÞVUÒ}�82§ü70/=s9'�s«q-¤�q/�=b�'/C(2�b�.9'¡��'=.-08q3'��8q3�=b�'/�4)(+=@('ç

i C ëHS T ¸Qìs»é'ºÚWùi C Wä¹:é ¸{á'$ ±_®:»

ò�ëHV�¸�¶ Û�é.¶ Ûm»}-0�}7¤8_¢n(+707¤£xM·-0�'�.¢G�'-0=.¼m¢8q3_=.-03_9'8:9'��°'ò�¸��<»�ûüºV¥{8q,_�ÛWåº'é�ò�¸��<»�û ¹�¥{8:,��Ûa ¹qé ¸{á'$ ±q±X»

¡�ë\V�¸Þ¶ Ûjé.¶ Ûm»>é0¡�¸��<» g º¨é"op��ëî¸Qº'é�¹n»>éìO¸��<»�û³�XWC ¡�¸ZY'»�Q[Y_é"op��ëî¬iº'é�¹G¯.é ¸{á'$ ±sJ:»c ë\V�¸�¶ ÛCéb¶ Ûm»}-0�s�.9'¢G�"=b�)(2= dfeÝg º¨é|cÚ¸��<»�a e é"o���ë"¶ Ûjé ¸{á'$ ±_²:»

­�ë¤V�¸�¶ Û�éb¶ Ûm»é v,�ë¤V�¸Þ¶ Û�éb¶ Ûm»I ¸{á'$ ±s[:»

²

Page 8: Modeling wells in porous media flows - CORE · Modeling wells in porous media ows Pierre Fabrie, Thierry Gallouet To cite this version: Pierre Fabrie, Thierry Gallouet. Modeling wells

ô�¨(ä7F�å�æ�å�çfF¯®E°MLèø"¶�µ�Ð=Ñ�Ò�émÔ>Ê]Ð=Ñ�Ò�Ó>ÑÔ�ê*Ð=Ñ�Ò�Ó�ë@ÔL¸(ÀfÅHÐ=Ñ�Ò�\>Ô�ê*Ð=Ñ�Ò�X]�Ôx·f´�ß�Å�Ò�^�Àf¶Ú¾¸�ý-¾uµ�·f¸(µs¸�iqéVI<»Ú¿�¾K¸z×0¶@¸�Ö�¾´�ß�Çsµ�¿�´(À´$ÂÚÐ1ÑsÒ?>�Þ�Ô�ê*Ð1ÑsÒ�sÑ�Ôx¿ ÂimëHS ? ¸.¸Qº'ébï�»U@ì 1 ¸{ìs».»>é�ºZWùi�Wÿ¹qéimëHS0T�¸b¸{º'ébï�»U@S}1n¸Qìmé.÷�»b»é�ºZWùi:¸Qè�é´ê@»�Wÿ¹qé«Â´�Ë}÷�ê$¸�Òö�Òsè�ë"ìmé«Â´�ËK¸(Ò�¶-Ò%êsëî¸Qº'ébï�»é ¸{á'$ ±_µ:»

i ð ë\S ? ¸.¸Qº'ébï�»U�¸1� 1 � � ¸{ìs».» � »é�o�� g Q)é ¸{á'$ ±st:»imëHV�¸.¬iº'ébï}¯$Un¸=� 1 � � ¸{ìs».» � »"op� g QEé�i:¸�I0é@ºq»�ûÁi C ¸$¿ÃÀz� 1 � � ¸Qìs».» � »>é ¸{á'$�Jqº:»

I�ëHS T ¸.¸Qº'ébï�»U�� 1 � � ¸Qìs».»é7op�më�¬0¹qé@áq»é ¸{á'$�J¨¹n»�ki ð ¸�I¤é´ê@»é@��� �¹î ï�ð�ñ � � � î ï�ð�ñ ñ � � õRìO¸�i�»¸Qè�é´ê@»�DnõR�}¸{èE»$Q_èmñ � � ­�¸�iq¸{è�éaê@»b»0D�õR��¸Qè)»�Q_èó � � æ�¸Qè�é´ê@»�DnõR�}¸{èE».ò�¸�iq¸{è�é´ê@».»$Q_è�ñ � � iq¸{è�éaê@»��}¸{èE».ö2¸Qè�é´ê@»$Qq÷�¸{èE»·ó � � �}¸{èE».úE¸{è�éaê@»�Qq÷�¸{èE»�ûüº'éop�ùë\�ä1 � ò ¸Qìs»é"opó g QEé|Â�´�ËK¸(Ò�¶�Òwê}ëî¸Qº'ébï�»é

¸{á'$�Jqá:»

æ H ¸{è�éaê@»%û ó]cÚ¸�iq¸{è�éaê@»b»<¸aõJI%¸{è�éaê@»·ó v,<¸�iq¸{è�é´ê@».»aþ�»·é«Â´�Ë�¸�Òö�Òm¸{è�é´ê@»�ë"ì|íî¸Qº'ébï�»éó � � æ H ¸{è�éaê@»�D�õRôs¸{èE»$Q_è�û � � ôs¸{èE».úE¸Qè�é´ê@»$Qq÷�¸{èE»�ó � � ôs¸{èE».ö+¸{è�éaê@»�Q:÷�¸Qè)»>é

o�ô�ë«�ä1 � ò ¸Qìs»é�o�ó g Q)é�´�Ë�¸�Òö�Òwê�ë�¸Qº'ébï�»I¸{á'$�Jq®:»

_�3¨4'/,�=b�'/�b/��_£X�E8:=b�'/�b/�n°E§}/x�b�)(2707<�',b82�:/:°¨�.//V�./¢=b-08:3\J¨°_=.�'/�¥{8q7¤7082§s-03'«m,./>�.9'70=nçõ�ö�¨sç�¬�¨s©÷®E°MLED�¾@¾Ç�»Ú¿�À�ÈÁÐ=Ñ�Ò�émÔ>ÊLÐ=ÑsÒ�[>�Ô>ê*Ð=ÑsÒ��]kÔ�¸(ÀfųÐ=Ñ�Ò�Ó>ÑÔ�ê*Ð=Ñ�Ò�Ó�ë@Ô�ÊLµ�·f¶-Ë$¶«¶@à(¿�¾µ�¾\¸(µÚß¹¶@¸�¾µR´(Àf¶h¾´�ß¹Çsµ�¿�´�Àè´�ÂÐ=Ñ�Ò?>(ÞÔ�ê*Ð=Ñ�Ò�ØÑÔx¿�ÀHµ�·f¶Z¾¶�Àؾm¶R´�Â�Å�¶�û0Àf¿�µ�¿�´�ÀÛÑ�Ò`>�Ò5s�'-0�s�.£_�b=./¡ -0�s�:/,b£��.-0¡�-¤7�(+,�=b8�=.�'/V¡�8_4'/7E�',.8q*'70/¡­°q/Á_¢/�¨=s¥{8q,s=.�'/m4'/«q/3'/,G(2=b/C4'- A<9'�.-08q3�=./>,.¡ 4'-0��¸�¡�¸�in»bõRin»>$�}9'=�=b�'-0��7�(2¢GÀj82¥�¢8_/,b¢-0�:/3'/>�.�s-0��¢>7k(+�.�b-¤¢�(27<(23'4�4'8_/�}3'8q=�70/n(24�=b8C(24'4¨-¤=b-08:3)(+7X4¨-Tâ�¢>9'70=.-0/��¢8q¡��)(+,./4�=b8�=.�¨/��',b8X8+¥¥{8q,s=.�'/m¡�8_4'/>7)�',b8:*'70/¡ª¸��b// à />¢=.-08:3HJq»$a b �� �� c��%���#då j� j�Je�� m�Gc ���>��������gf��� m�����¶a3O=.�¨-¤�V�./¢=b-08:3O§}/��',.82�q/�/Á_-¤�b=./3¨¢/:°�9¨3'-0ÅX9¨/3'/�b� (+3'4­�b=@(+*'-07¤-0=a£",b/�.9¨7¤=b��¥Q8:,m=.�'/�/>7¤70-0�'=b-¤¢j�',b8:*'70/¡ª-03_�:8:70�q/4"-03¸{á¨$0¹n»��¸{á¨$�J:»$·¶a3'4'//4�°)=b�'/m�',.8q*'70/¡ =.8�*E/x¢>8:3'�b-¤4¨/,./4�-¤�

ós4'-0� ' ýC¸{õmßjóA;�» ( ¸{èE»%ûå÷�¸Qè)»>és-03�ìméýC¸Qè)»>¸{õmß�¸{èE»�óh;�¸{èE».»�D@F­ûüº'é}-03HG)ìmé ¸{®'$0¹�»

9'3'4¨/,�=.�¨/j¥{8:707082§s-03'«C�_£_�E8:=.�¨/�./�j8:3�=.�¨/C4)(+=@(�¸�,b/¢n(+7¤7w=.�)(+=lc P ¸Þ¶ Û�»�4'/3¨8:=./>��=.�'/��b/=�8+¥½Q�í\Q�¡�(+=.,b-¤¢>/�m§s-¤=b�¢8_/Gâ�¢-0/3_=b��-03�¶ Ûm»ç

ý�-0�]bs(+,@(2=b�'/8_4'8q,.£�¥Q9'3'¢=b-¤8q3�¥{,.8q¡ ìwêVi�c P ¸�¶ ÛV»}�@(2=b-0�a¥{£_-03'«dfeÝg ºj�.9¨¢@��=.�)(+=sýC¸{èE»�jKD@jÚa e jKD@j)é"ofj�ë�¶ Û P éw¥{8:,�(¨$ /q$�è ë ìmédsr ë�¶ Û �.9'¢G�"=b�)(2= n ýC¸Qè)» n W r é}¥Q8:,�('$ /:$�è�ë"ìmé ¸{®'$iáq»

;ÿû ¸Z; 1 é-D�D-DGéV; P » ð ëî¸�S�?:¸Qìs».» P é÷­ëz�Ò¸{ìs»>é%÷�¸{ìs»�ûüºfI ¸{®'$i®q»

Ûs/>¢n(2707<=.�E(2=�FO4'/3¨8:=./>�x=b�'/m8:9'=a§s(2,b4�9'3'-0=�3¨8:,.¡�(27E�:/¢>=.8:,�=.8�ìå8q3HG)ìm$[

Page 9: Modeling wells in porous media flows - CORE · Modeling wells in porous media ows Pierre Fabrie, Thierry Gallouet To cite this version: Pierre Fabrie, Thierry Gallouet. Modeling wells

jsÎbÏ k�l����4�'Ô��g�<Ôm�!��Óong����pqn�Ô���Ô��[�r 3'/mÄ),b�.=s�',b82�:/�m(�«q/3'/,G(27�/Á_-¤�b=./3¨¢/C,b/�.9¨7¤=m(2�s-03­¬�J>¯�(23'4�¬¤¹-J>¯.$��¬�ç��0ç�{å�æ�å�ç�FsLE°�±ut vqw�å1{æ-¨sFx��¨Jy$ç�¬Ræ�ö�¨z¨[ Z �å��|æ-åZ�Z¨�z�{|ªqæ�å�ç�F:|ø"¶�µ�Ð)>�Ò�Ñ�Ô>ê*Ð)>sÒ?>Ôu·f´�ß¹Å(ÒXÙ�·q¶�Ë�¶K¶@à�¿�¾µÃ¾K¸R¾´�ß�Ç�µ�¿�´(À�ß«µ�´zÐ)>�Ò�Ó*Ô`¿ÃÀHµ¹·q¶¢Â�´�ß�ß�´�×0¿�À(ÈÚ¾¶�Às¾¶��ß�ë«� 1 � � ¸Qìs»é�o��mëO¬¤¹qé QQmó ¹ »>é � � ß�¸Qè)»�Q_è�ûÿº¨é� � ýj¸{èE».õmß�¸Qè)»�DõR�}¸Qè)»�QXè�û � � ýj¸{èE»�;�¸Qè)»"DnõR�}¸Qè)»�QXèmñ � � �}¸Qè)»�Q:÷�¸Qè)»>é�o��Oë~}ò�� P � 1 � ò ¸{ìs»I

¸{®'$ ±_»

��¬�ç�ç2y]ç�y��l¬�ç��0ç�{�å�æ-å�ç�FAL�°�±5s�'/m�',b8X8+¥��',b8_¢//4'��-03�=a§}8j�.=./>�'��(23'4�-0�}�:/,b£��.-0¡�-¤7�(+,�=b8C=b�)(2=s8+¥e¬�J>¯.$�¶a3 à =b/�O¹:°_�b8:¡�/s*E8:9¨3'4'�s8:3�=b�'/m9'3'-0Å_9'/�.8q709'=.-08:3j82¥�(23�(+�'�',b82ÁX-0¡�(2=./%�',b8:*'70/¡�(+,./s«:-0�q/3�$�¶a3 à =b/��ᨰ2�)(+�.�.-03'«V=.8m=.�'/�70-¤¡�-0=<-¤3j=.�¨/�(2�'�',b82Á_-¤¡�(2=b/��',.8q*'70/¡�.9¨�'�'70-0/�s=.�'/V/Á_-¤�b=./3¨¢/�,./>�.9'70=n$��æ�¨[�̱�°<t �J�G��¬�ç�w�å�©«ªfæ�¨#��¬�ç��G �¨s©m|M</>=Éò�-03HS�?q¸{ìs»�*E/C�b9'¢G�­=b�)(2= � � ò�¸{èE»$Q_è"û º'$� �b=.,@(+-0«:�_=a¥{8q,.§s(2,b4­(2�'�¨7¤-0¢n(+=.-08:3�8+¥½=b�'/RM�(2Á(�.à�-070«:,@(+¡�M</>¡�¡�(70/n(24¨�s=.8�=.�¨/x/>ÁX-0�b=./3'¢>/ (+3'4�9'3'-0Å_9'/3'/�b��8+¥½(j�.8q709'=.-08:3jß�8+¥

ß�ëHì 1 ¸Qìs»é � � ß�¸{èE»$Q_è�û庨é� � ýC¸{èE».õmß�¸{èE».D�õR�}¸{èE»$Q_è�û � � ýj¸{èE»�;�¸Qè)»�D�õR�}¸{èE»$Q_èmñ � � �}¸Qè)»bò�¸Qè)»�QXè�é�o��OëHì 1 ¸Qìs»I ¸{®'$�Jq»}�8q=./�=b�)(2=�§s-0=b�'8:9'=m=b�'/��_£X�E8:=b�'/�b-¤� � � ò�¸{èE»$Q_èOû º�=.�'/xM�(2Á(�´à -¤70«q,@(2¡¥M</>¡�¡�(j«q-0�:/�m=.�¨/�/Á_-¤�b=./3¨¢/"(23'4�=.�'/9'3'-0Å_9'/3¨/�.��82¥½(j�b8:709'=b-¤8q3Cß�8+¥e¸{®¨$�J:»w*'9'=s§s-0=.�H�ùë¤ì�1�¸{ìs»}�.9¨¢@��=.�)(+=]� � �}¸Qè)»�QXè�ûüºj-03'�.=b/n(24�8+¥7�ùë¤ì�12¸{ìs»>$ÂüÄ),b�.=�/>�.=.-0¡�(2=./�8:3�=.�¨/:°'�b8:709'=.-08q3Cß�82¥�¸{®¨$ Jq»�-0�s8q*'=@(+-¤3¨/4�*_£�=G(2À_-03'«R�­ûù�X�E¸Tß)»>°_§s-0=.�C=�ë"¶ Û O §s�'/>,./

� � ¸�i�»�ûäó�¹qé�i�W|ó�= ó ¹:é� � ¸�i�»�û³i}ñu=·é<ó�=Vó ¹g�ùiT�üó�=<é� � ¸�i�»�ûÿº¨éEó�=�W�ilW�=<é� � ¸�i�»�û³i�ós=·é�=���i���=mñå¹qé� � ¸�i�»�û ¹qé�=mñÿ¹YWùi�I&¨,.8:¡ (�¢7�(2�b�.-0¢n(27%,b/�.9'70=m82¥ à =@(+¡��)(+¢¢G�'-�('°��X�E¸Tß)»�ë¯ì�12¸{ìs»�(23'4OõR�X�'¸TßE»sû �"�� ¸ãß)».õmß�°­(qI /(I"8:3�ìm$��å-0=.�"=b�'-0�¢G�'8:-0¢/m8+¥��ù-03­¸{®¨$�J:»°_8q3'/m«:/=b�

� ��� n õmß�¸Qè)» n ? W�V 1 ñ áe#� ò ��� ï��B��� é ¸{®'$i²q»

§s�'/,b/uV 1 4'/�E/3'4'�m8q3'70£�8:3 e ° � ; � ? ° � ý � T ¸ � ; � ? 4¨/3'8:=b/��=b�'/uS�?q¸{ìs»k� 3¨8:,.¡ 82¥ n ; n (+3'4 � ý � T =.�'/KS0T�¸{ìs»��3'8q,.¡ 82¥ n ý n »}(23¨4�§s�¨/,./ �q��û~�2è�ë ìmé�=`W n ß�¸{èE» n W�=mñå¹���I6m-0�:/>3L����QE°q¥{,b8:¡�=b�'/ à 8:*E8:70/��-03'/Å_9)(270-0=a£_°2=.�'/>,./j/Á_-¤�b=.�]V ? 4'/�E/3'4¨-¤3¨«�8:3¨7¤£�8q3�ìü(23'4H�V�.9'¢G�"=b�)(2=

�/� ó v��� � @�� �B�p� W V ? � õ �T� � � @m�B�p�Ã��� é"o � ë«� 1 � � ¸{ìs»>é ¸{®'$�[q»§s�'/,b/ v� -0��=b�'/�¡�/n(23��:(2709'/�8+¥ � 82�q/,�ìm°f� N ûÁ��Q[�'¸�Q�ó«�2»s(23'4

� õ ��� � @ �B�p� � û ¸ � � n õ � ¸Qè)» n � Q_èE» ï@ I_��b-03'«­¸Q®'$i²:»�(23'4�¸Q®'$�[:»V¸�§s-0=.� � ûåß)»�§�/j�.�E(2707%3'82§ä�',.82�q/�=.�E(2=n°¨¥{8q,�(2707�¹ZWü�J�èQ��¨¸�QÉóÿ¹n»>°)=b�'/,b/�/Á_-¤�b=.�KV:�:°4'/�E/3'4¨-¤3¨«�8:3'70£�8q3 e ° � ; � ? ° � ý � T °'ìm° � ò ��� ï��B��� (+3'4L�_°¨�.9'¢G��=.�)(+=

µ

Page 10: Modeling wells in porous media flows - CORE · Modeling wells in porous media ows Pierre Fabrie, Thierry Gallouet To cite this version: Pierre Fabrie, Thierry Gallouet. Modeling wells

� ß � î�ï�ð @� �p� W�V � I ¸{®'$iµq»&¨9',.=b�'/,.¡�8:,b/lV:�m¡�(�£C*E/m¢@�¨8:�./>3"=.8�*E/m3'8:3'4¨/¢,./�(2�.-03'«�§s-0=.��,./�b�)/>¢=m=.8 � ò ��� ï �B��� ° � ý � T (23'4 � ; � ? $¶a3�8:,b4'/,�=b8��',.82�q/�¸{®'$iµq»°'§}/�(2�b�.9'¡�/ ¹lW����ùQ��¨¸�QÉó ¹�»$ à -03'¢/K���³Q�(23'4��.-03'¢>/C=b�'/m¡�/n(+3��:(2709'/m8+¥·ß�82�q/,�ì-0��º'°<¸Q®'$�[:»w«:-0�:/>�

� ß � �� @�� �B�p� W ¸�V ? » � � � n õmß�¸Qè)» n � Q_è�ûä¸�V ? » � T���7�C � ��� n õmß�¸Qè)» n � Q_è"I ¸{®'$�tq»Ã -03'¢/K���åá'°_9'�b-¤3¨«�=.�'/T�C;8q704'/,s-03'/Å_9)(+7¤-0=a£�-03­¸Q®'$�t:»w70/n(24¨�s=.8

� ß � �� @����B��� Wä¸�V ? » � T���7�C ¸�V 1 � ò ��� ï$�B��� » @� ¸�¡�/�(2�n¸�� � ».» 1 � @� IM</>=]V: �û ¸�V ? » � ¸�V 1 � ò � � ï@�B��� » @� ¸Þ¡�/n(+�¸{ìs»b»G1 � @� (+3'4HVq¡sû ¸�V ? » � ¸�V 1 � ò � � ï$� �p� » @ � $�}�8:=b-¤3¨«C=b�)(2== � � ¡�/�(2�n¸�� � »0W � � � n ß�¸Qè)» n � � Q_è�é=.�¨/x�¨,./�_-08:9'��-03'/Å_9)(270-0=a£j£X-0/704'��°_¥{8q,e(+3_£<¢Haä¹:°

� ß � �� @ � �B��� W�¢�V: �ñ¯Vq¡ T���7�£ ¸ ¹= » �*@�¤�*¥ @*¦� ¤ � ¥ @*¦A§ � � � n ß�¸{èE» n � � Q_è�¨ 1 �

@�(23¨4�°_9'�.-03'«�8q3'¢/m¡�8q,./�=.�'/©�j;8:704'/,s-03'/>ÅX9E(270-0=a£X°

¸ � � n ß�¸Qè)» n � � QXèE» @@�� û � ß � �� @����B��� Wª¢�V: �ñ Vq¡:¸ � � n ß�¸Qè)» n � � Q_è)» �*¥ @� T���7�£ ¸ ¹= » ��¤�*¥ @*¦� ¥ @ I ¸{®'$0¹�º:»

à -03'¢/H�«� Q[�'¸�Q�ó ¹�»°�=b�'/3³QE¸Qájó��2»��'¸�Q�ó��2» g ¹"(+3'4�=b�'/"�b/,.-0/"-03ÿ¸Q®'$0¹nºq»�-0��¢8q3_�:/,b«:/3_=n$å&¨9',b=.�'/,b¡�8q,./:°����� N û ¸�Qjó �2»���Q¬��¸{áVó �2»��qá�-T¥]Q�û�®"(+3'4 ���(� N û�¸�Q ó �2»��(Q�û ¸{áÉó��2»��:á�-㥽Q�û�á¨$�¶Q¥]Q"û�®­¢�û�¹q°�-ã¥Q�ûÚá�8:3'/j¢G�'8_8:�./>�®¢O�b9'¢G�­=b�)(2=KV ¡�¯ T��7�£ ¹/�4='?ÚW�¹/�qá'$�5s�'/3�°�=.�'/j-03'/Å_9)(+7¤-0=a£�¸{®'$0¹�º:»s«q-0�:/�m=.�¨/C/Á_-0�.=b/3'¢/�8+¥Vq°2°¨4'/�E/3'4'-03'«�8q3'70£�8:3 e ° � ý � T ° � ; � ? °'ìm° � ò ��� ï*� �p� (+3'4H�_°_�.9'¢G�"=b�)(2=� ß � � @ � � �p� W�Vq°�I5s�'/,b/G¥{8q,./:°%«q8:-03'«�*E(2¢GÀ­=b8O¸{®'$�tq»�=b�'/,b/�/Á_-0�.=b�uV:�:°�4'/�E/3'4'-03'«"8q3'70£"8:3 e ° � ý � T ° � ; � ? °�ìm° � ò ��� ï@�B��� (23¨4 �_°�.9¨¢@��=.�)(+=�¸{®'$iµq»��'8q704'�n$%5s�'-0�}¢8:¡��'70/=./>� à =./�O¹q$��æ�¨[�~®�°<t �Yª�{�{�å�F�þzæ-ç\æ-ö�¨­ �å�©«å�æ4|5s�'/,b/�/Á_-0�.=b�x(V�./Å_9'/3¨¢/�¸Qò £ » £2±�² ³ � S0T�¸{ìs»}�b9'¢G�"=.�)(+=¹q$sò�£s´�÷�¥{8q,m=.�'/�§}/n(2À(��µ­=.8q�)8q708:«:£�8+¥É¸�V�¸ ìs»b»1�s(2�®¢«´ â ¸�-a$ /:$ � � ò�£�¸{èE»$�}¸{èE»$Q_èA´ � � �}¸{èE»$Qq÷�¸Qè)»>°�(+�¢<´ âü°q¥{8q,�(2707p�ùëHV�¸ ìs»b»°á¨$ � ò £ ��� ï �B��� W n ÷ n ¸Qìs»%¥{8q,�(2707�¢�ë�¶ }�°®¨$ � � ò £ ¸{èE»$Q_è�ûÿºÉ¥{8q,�(2707K¢�ë�¶ }�$

Ûs/>¢n(2707�=b�)(2= n ÷ n û ÷ O ñå÷G��°�§s�'/,b/�÷ O (+3'4O÷G�ÿ(2,b/�=.�'/�¢7�(2�.�b-0¢n(27j¾.�E8:�b-¤=b-0�:/¿"(23¨4ä¾.3'/«_(2=b-¤�q/¿��)(2,b=.�j82¥s=.�'/¡�/�(2�.9¨,./m÷ ¸Þ�.8�=b�)(2=s÷�ûå÷ O ó­÷G��»°'(+3'4�=.�)(+= n ÷ n ¸Qìs»�-0��(+7¤�b8C=b�'/m3'8:,b¡ 8+¥�÷�-03�=b�'/m=.8q�)8q7¤8q«:-0¢n(+7)4'9E(27<�.�E(2¢/�8+¥V�¸ ìs»}/3'4'82§}/4�§s-0=.��=.�'/KS0T�¸{ìs»k� 3'8q,.¡�$&¨8:,�¢�ë�¶ }�°270/=�ß�£j*)/s=b�'/s�.8q7¤9¨=.-08:3j82¥�¸{®¨$ Jq»�§s-¤=b� ò�£j-03'�.=b/n(24�8+¥·ò�$ à =b/�"¹s�.�¨82§s��=.�)(+=%=.�'/s�b/Å_9'/3'¢/�¸ãß�£)»)£�±�² ³�-0�*E8:9'3'4¨/4�-¤3z�ä1 � � ¸Qìs»%¥Q8:,s(2707�¹YWù�T�ùQ[�'¸�Q�óO¹n»$%5s�¨/3�°_8:3¨/�¡�(�£j(2�.�b9'¡�/q°_9'��=b8 (m�b9'*'�b/Å_9'/3'¢/q°<=.�)(+=�ß2£¶´ ß-03H�ä1 � � ¸Qìs»w¥{8:,s=b�'/m§�/n(+À�=b8:�E8:708:«q£�¥{8q,�(2707%¹�W ����Q[�¨¸�Q�ó ¹�»$%5s�'/>,./G¥{8q,./:°E�)(+�.�b-¤3¨«C=b8�=.�'/m70-0¡�-0=�(2�q¢­´ âL-03¸{®¨$�J:»m¸�§s-0=b��ß�£­(+3'4­ò�£"-03'�b=./n(+4"82¥%ß­(23¨4­ò)»�8q3'/�«:/>=.�m=.�)(+=�ß�-¤�m(��b8:709'=.-08q3�8+¥x¸{®¨$ ±X»$�¶a3"8:,b4'/,m=.8��)(2�b��=.8�=.�'/70-0¡�-0=%-¤3�=b�'/m,.-0«:�_=s�)(+3'4��.-04'/m82¥s¸{®'$�Jq»°_3'8q=./m=.�)(+=��ä1 � ò ¸{ìs»��ùV�¸ ì�»>°'-ã¥7ó g QE$%5s�'-0�s¢8q3'¢709'4'/� à =./>�­áC(+3'4�=.�'/�',b8X8+¥�82¥½#½,.8:�E8:�b-0=.-08:3�®'$0¹:$

t

Page 11: Modeling wells in porous media flows - CORE · Modeling wells in porous media ows Pierre Fabrie, Thierry Gallouet To cite this version: Pierre Fabrie, Thierry Gallouet. Modeling wells

§R¨s©«ªf¬�­uLE°�±¸·�Ë�´@¼f´�¾¿�µ�¿�´�À->sÒ�Ó�¿�¾RÏ�Çs¿�µ�¶KÈ�¶-Àf¶-Ë$¸�ß0¾¿�Àfº@¶R¿�µ0È(¿�É�¶�¾R¸�À«¶@à�¿�¾µ�¶�Àqº*¶�Ë$¶�¾Ç�ß�µ�´�ËÚ¸�ÀqýL»Ú¶@¸�¾ÇsË$¶V÷ù¾Çsº*·µ�·f¸�µw÷�¸{ìs»�û ºqÒx²K´(µ�¶�¸�ß ¾m´zµ�·f¸(µ]µ�·f¿�¾Y¼pË�´�´�Âu»Ú¸�ýzÕ@¶x¸(Å�¸@¼�µ�¶@Åhµ�´�È�¿�É-¶R¶@à�¿�¾µ�¶�Àqº*¶�´(ËuÀq´�Àsß�¿�Àf¶@¸�ËR´m¼Ø¶-Ë$¸(µ�´(Ë@¾K¾Çsº*·¸�¾Kø"¶�Ë�¸�ý-ê�ø0¿Ã´�Às¾x´m¼Ø¶-Ë$¸(µ�´(Ë@¾u¸�¾x¿ÃÀCS1ë�UÒ~Ù�·f¶`¾´�ß¹Çsµ�¿�´�ÀhÈ�¿ÃÉ�¶-À«¿�À¹·½Ë$´@¼f´�¾¿�µ�¿�´(À->�Ò�Ó>Ê]µ¹·q¸�µ�¿�¾Rµ¹·q¶R¾´�ß�Ç�µ�¿�´(À«´$ÂLÐ)>sÒ�-Ô¿�¾xÇsÀf¿�Ï-Çs¶H¿ ÂÚQOû�á êuµ¹·q¿�¾x×0¿�ßÃߢÕ@¶Ú¼pË�´�É-¶�À~¿�À µ¹·q¶HÀf¶@à�µ�¼�Ë$´m¼Ø´�¾m¿Ãµ�¿�´(À«ê1ÊKÇsÀ´(˵�ÇsÀf¸(µ�¶�ß¹ýhµ�·f¿�¾xÇsÀf¿�Ï-Çs¶�Àq¶¾@¾HË$¶�¾Ç�ß�µ¿�¾uÀq´zß¹´(À�È�¶-ËÚµ�ËÇs¶R¿ Â�Q�û ®fÒJDͺ@´�ÇsÀfµ�¶�Ë�¶*à(¸�»½¼�ß�¶xµ�´LÇsÀf¿ÃÏ�Çs¶-Àf¶¾@¾Rº@¸�À«Õ@¶xº@´�Àؾmµ�ËÇsº-µ�¶@Åz¸�¾R¿�À­S$Ó���UR×.·f¿�º@·Ú´�ßÃß¹´�×7¾¶�¾*¾¶-Àfµ�¿�¸�ß�ß�ýxµ�·f¶R×0´(Ë$Öz´$ÂZþ�¶-ËË¿�À­S$Ó�é�UÒ�º*À«´�Ë�Å�¶-Ëuµ�´z¸�ß ¾m´�·f¸�É-¶x¸(À\¶@à(¿�¾µ�¶-Àfº@¶x¸(ÀfÅHÇsÀf¿�Ï-Çs¶�Àf¶�¾@¾xË$¶�¾Ç�ß�µ�Â�´�Ë�Q�û ®ØÊ×0¶Y¾$·f¸�ß�ß.Ç(¾m¶Z¸�ÀH¸(Å�Å�¿Ãµ�¿�´(Àf¸�ß.¸�¾@¾mÇs»½¼pµ�¿�´�ÀØÊ7¿ÃÒ�¶�Ò�Ð1ÑsÒ�Ó�ëmÔ�ÒuÙ�·f¿�¾<¿�¾]Å(´�Àf¶Ú¿�ÀJ·�Ë$´m¼Ø´�¾¿�µ�¿Ã´�ÀJ>sÒ�ÑxÕ@¶ß�´�×0Ò:D�Àf´�µ�·f¶-Ë�×0¸�ýÚµ�´´�Õ-µ�¸�¿�ÀR¶@à�¿�¾mµ�¶-Àfº@¶]¸�ÀqÅKÇsÀf¿�Ï-Ç�¶-Àf¶�¾@¾.´�Ë�¸�Àfý�»Ú¶@¸�¾ÇsË$¶%÷«×0´�Ç(ß�Å<Õ@¶Yµ�´�Ç(¾¶]¸uOÃÅ�Çs¸�ß�¿�µ�ý�»Ú¶�µ�·f´-Å�P½¸�¾�¿�ÀKþ�µ�¸(»�¼f¸�º@º@·f¿�¸X» ¾×0´(Ë$Ö©S$Ó�¼�U@Ê¢×"·f¿�º@·z×0´�Ç(ß�ÅRß¹¶@¸(Åxµ�´R¾´�ß�Ç�µ�¿�´(Às¾K´$ÂuÐ)>�Ò�Ó�Ôx¿�ÀH¸EO�¾µ�Ë�´�À�È(¶-Ë]¾¶-Às¾¶�PÚµ�·f¸(ÀHµ¹·q¸�µ�´$ÂuÐ)>sÒ�-Ô�Ò��¬�ç��0ç�{å�æ�å�ç�FsLE°¹®~vqw�å1{�æ�¨sF���¨hªfF�½E{�F|å*z�{�¨sF|¨f{-{Ty$ç�¬zæ�ö�¨«¨[ � �å���æ�åZ�`¨�z�{|ªqæ�å�ç�Fùø"¶�µ�Ð)>�Ò�Ñ�Ô>ê*Ð)>sÒ?>ÔH¸�ÀfÅhÐ1ÑsÒ�Ó�ëmÔ·f´�ß¹ÅzÐ$Àq´�µ�¶Rµ¹·q¸�µ�Ð1ÑsÒ�Ó�ëmÔx¿�¾K¸�ß¹×�¸�ý-¾Kµ�Ë�Ç�¶Ú¿�ÀHµ�·f¶Úº@¸�¾¶KQ�ûäá>Ô�ÒRÙ�·q¶�Ë�¶u¶@à�¿�¾mµÃ¾K¸xÇsÀf¿�Ï-Çs¶Ú¾´�ß¹Çsµ�¿�´�ÀV߯µ�´LÐ)>�Ò�Ó*Ôx¿�À\µ�·f¶¾¶�Àؾm¶Úµ�·f¸�µ:�ß�ë\� 1 � � ¸Qìs»é"op�VëO¬0¹:éGá:»é � � ß�¸Qè)»�Q_è�ûÿº¨é� � ýC¸Qè)»bõmß�¸Qè)»�D�õR��¸Qè)»�Q_èCû � � ýj¸{èE»�;�¸Qè)»�D�õR��¸Qè)»�Q_èxñ � � �}¸{èE»$Qq÷�¸Qè)»>éop�Oë~}ò�� P � 1 � ò ¸{ìs»�I

¸{®'$0¹q¹n»

§R¨s©«ªf¬�­uLE°¹®Ó�Ò�ø.¶-µ�÷ ë~�Ò¸{ìs»R¸(ÀfÅ�¹xW ó�� âÁÒ¯Ù�·f¶z»Ú¶*¸�¾Ç�Ë�¶�÷èÕ@¶ß�´�À�È�¾Rµ�´"¸=�ä1 � ò ¸{ìs»b»1�K¿ ÂÚ¸�ÀfÅh´�ÀØß¹ýH¿ ÂÚµ�·f¶-Ë$¶x¶@à(¿�¾µ�¾¾m´(»u¶ÚË$¶@¸�ß.ÀfÇs»uÕ@¶-Ë]Vÿ¾mÇsº@·hµ¹·q¸�µ

� � �}¸{èE»$Qq÷�¸{èE»�W�V � � � î�ï�ð�ñ$�B��� é"op�ùë\� 1 � ò ¸Qìs»��LV�¸ ìs»I ¸{®'$0¹�á:»

º*ÀfÅ(¶*¶@Å�Ê¢¿ Â�µ¹·q¶u»Ú¶@¸�¾ÇsË$¶s÷~¾¸(µ�¿�¾�û0¶�¾uÐ)>�Ò�Ó>ÑÔ`Р´(Ë�¾´�»Ú¶]óVëî¬0¹qé*â »Y¸�ÀfÅ`¾m´(»u¶�V ë"¶ Û½Ô>Ê�÷ º@¸�ÀÛÕ*¶ÚÇsÀf¿�Ï-Çs¶ß�ý¶*à(µ�¶-ÀfÅhµ�´Û¸zß¹¿�Àq¶*¸(ËRº@´�Àfµ�¿�ÀfÇs´�Ç(¾x¸m¼�¼�ß�¿�º@¸�µ�¿�´�ÀØÊ�¾mµ�¿�ß�ß0Å(¶�Àf´(µ�¶@ÅhÕ�ý�÷�Ê.ÂË$´(»Í�ä1 � ò ¸{ìs»�µ�´�¶ ÛuÒ¿¾|Àq¿�Ï�Çs¶-Àf¶�¾*¾x´�µ¹·q¶u¶@à�µ�¶-Às¾¿�´�ÀKÂ�´�ß�ß�´�×�¾�ÂË$´(»�´$ÂKµ�·f¶ÚÅ�¶-Às¾¿�µ�ýR´$Â��ä1 � ò ¸Qìs»��zV�¸ ìs»Z¿�Àz�ä1 � ò ¸{ìs»�ÒÑ�Ò�ø.¶-µKµ�·f¶x·qý�¼Ø´(µ¹·q¶¾¶�¾L´$ÂJ·�Ë�´@¼f´�¾¿�µ�¿�´�À«>sÒ�Ñ\·f´�ß¹Å-À�ÂË$´(» ¸«Å(¶�Àؾm¿Ãµ�ýh¸�Ë�È�Çs»Ú¶�ÀfµÃÊuµ�·f¶KÂ�´�Ë»ÚÇ(ß¹¸�µ�¿�´(À�Ð)>sÒ�Ó�Ó*Ô ¿�¾¶*Ï-Çs¿�É-¸�ß¹¶-Àfµ�µ�´�µ¹·q¶¢Â�´�ß�ß�´�×0¿�À(ÈR´�Àf¶��

ß�ëh� 1 � � ¸{ìs»é"o���ëO¬0¹:éGá:»>é � � ß�¸{èE»$Q_è�ûÿº¨é� � ýj¸{èE».õmß�¸Qè)»�D�õR�}¸{èE»$Q_è�û � � ýj¸{èE»�;�¸Qè)»�D�õR�}¸{èE»$Q_èmñù�Q÷�é*��� �¹î�ï�ð6Á1� � ����î�ï�ð6Á1� éop�ùëh�ä1 � í ¸Qìs»é�o�i g áfI¸{®'$0¹�®:»

Âx¶Z¾=·q¸�ß�ß�Å�¿�Ë�¶@º�µÃß¹ý½¼pË�´�É-¶KÕ@¶ß�´�×ùµ�·f¶Ú¶*à(¿�¾µ�¶�Àfº@¶Ú¸�ÀfÅzÇsÀf¿�Ï-Çs¶�Àq¶¾@¾u´�Â<µ�·f¶Z¾´�ß�Ç�µ�¿�´(ÀH´$ÂuÐ)>sÒ�Ó�>�Ô�Ò>�Ò�²K´�µ�¶Kµ�·f¸(µ�Ê�Â�´�Ë�Q�ûü®sÊ0µ¹·q¶u¶@à(¿�¾µ�¶-Àfº@¶uË�¶¾Ç(ß¹µ�¿�À<·�Ë�´@¼f´�¾¿�µ�¿�´�À�>�Ò�ÑL¿�¾�Àf´�µ�Å�¿�Ë�¶@º�µÃß¹ý�È�¿ÃÉ�¶-ÀHÕ-ý©·�Ë$´m¼Ø´�¾m¿Ãµ�¿�´(À­>�Ò�Ó¾m¿ÃÀfº@¶u¿�À�·�Ë$´m¼Ø´�¾m¿Ãµ�¿�´(ÀC>sÒ�ÓRµ�·f¶Z¾´�ß�Ç�µ�¿�´(À\Õ@¶�ß¹´�À(È�¾Kµ�´x�ä1 � � ¸{ìs»�´(ËK¸�ßÃß��jë�¬0¹:éG®0�:áq»]¸(ÀfÅz¿�À<·�Ë�´@¼f´�¾¿�µ�¿�´�À¬>�Ò�ѵ¹·q¶K¾´�ß�Çsµ�¿�´(ÀHÕ@¶ß�´�À�È�¾Kµ�´`�ä1 � � ¸Qìs»�´�Ë�¸�ßÃß���ëO¬0¹:éGá:»�Ò��¬�ç�ç2y]ç�y��l¬�ç��0ç�{�å�æ-å�ç�FAL�°¹®5s�'-0��-0�m�',b82�:/3"*_£�9'�b-¤3¨« (�¡�8_4'-ãÄ)¢n(+=.-08:3�82¥e(�,./«q9'7�(2,b-¤=a£�,b/�.9'70=m4'9'/j=.8�à�/£q/,.��¬0¹�á>¯½-03"=b�'/j¢n(2�b/C8+¥Ã%�-¤,b-0¢@�¨7¤/>=*E8:9'3'4E(2,.£O¢>8:3'4'-0=b-¤8q3'��¸{(23¨4 (2�b�.9'¡�-03'«"=.�E(2=j=.�'/�*E8:9'3'4E(2,.£O8+¥ÉìÚ-0�j�.¡�8X8q=.�)»>$h�O/�9'�b/��¨/,./"(��:(2,.-�(+3X=V§s-¤=b�}�/>9'¡�(+3'3�*E8:9'3¨4)(2,b£�¢8q3'4'-0=.-08:3�=.�)(+=s¢n(23�*E/m¥{8q9'3'4�-03�¬¤¹�®�¯�(+3'4"(270�.8j-03­¬�t�¯.$%�/GÄ)3'/

ì 1N ¸Qìs»�ûÄ�³Ü�ë\ì 1 ¸{ìs»�U � � Ü�¸{èE»$Q_è�ûÿº\��I&¨8:,sò"ëO¸�ìî1³¸Qìs».»1�a°_*_£�=b�'/m¢7�(2�b�.-0¢n(27�M�(+Á��.à�-070«:,G(2¡�M</¡�¡�(¨°n=b�'/,b/�/Á_-0�.=.�m(j9'3'-0Å_9'/�Ü��.9'¢G�"=b�)(2=

¹nº

Page 12: Modeling wells in porous media flows - CORE · Modeling wells in porous media ows Pierre Fabrie, Thierry Gallouet To cite this version: Pierre Fabrie, Thierry Gallouet. Modeling wells

Ü�ë¤ì 1N ¸Qìs»é� � õ�Ü�¸{èE».D>ýC¸Qè)»bõR��¸Qè)»�Q_è�ûè�aò)é@��� � ï � � � ï é"op�ùëHì 1N ¸Qìs» ¸{®'$0¹>±X»

(23¨4O=.�'/�8:�E/,@(+=.8:,jïÚ¥{,b8:¡L¸�ì�12¸{ìs»b»1�}=.8«ì�1n¸{ìs»V4'/GÄE3'/4�*_£�ï�¸{òE»�û�Üå¸Þ§s�'/,./�Ü�-¤�V=.�'/�9'3'-0Å_9'/��.8q7¤9¨=.-08:3O=b8¸{®¨$0¹±X»b»�-0�s70-03'/n(+,e(+3'4�¢8:3_=.-03_9'8q9'�n$5s�'/m,./>«:9'7�(2,b-0=a£�,./>�.9'70=s-03O¬¤¹�®>¯�«:-0�:/>�s=.�'/m/Á_-0�.=b/3'¢/j82¥0ó�C g á'°4'/�E/3'4¨-¤3¨«�8:3'70£�8:3�ý (23'4�ìm°_�.9'¢G��=.�E(2=�Ü�*)/>7¤8q3'«:�s=b8x�ä1 � ò ¸Qìs»�-ã¥�ò�*)/>7¤8q3'«:�s=b8�¸=�ä1 � ò � ¸Qìs».»1��§s-0=.��á��ùó�� ó�C(23¨4Ló-�)ûùó��¨¸�ówóO¹n»�¸�3'8q=./�=b�)(2=m¸1�ä1 � ò � ¸{ìs»b»1�"� ¸�ì�1n¸{ìs»b»1��»$�&¨9',b=.�'/,b¡�8q,./:°q¥{8q,�(23_£�á#� ó��ùó�C2°q=.�¨/x8q�)/>,@(2=b8:,}ï ò¸�4¨/GÄ)3'/4O*_£�ï ò ¸{òE»�ûÚÜ�§s�'/,./�Ü�-¤�m=b�'/C9¨3'-0ÅX9¨/��.8q7¤9¨=.-08:3�=.8O¸Q®'$0¹±_».»}-0��70-03'/n(+,.70£�¢8:3_=.-03_9'8q9'�s¥{,b8:¡Ò¸=�ä1 � ò � ¸{ìs»b»1�=.8z�ä1 � ò ¸{ìs»>$�¶a=s-0�}-¤¡��)8q,.=G(23_=�=b8C3¨8:=.-0¢/V=.�)(+=�=b�'/�,@(+3'«:/m8+¥�ï ò °_4'/>3'8:=b/4"*_£&Åj¸�ï ò »>°'-0��� 1 � òN ¸Qìs»s§s�'/>,./-03

� 1 � òN ¸{ìs»wûE��Æjë«� 1 � ò ¸{ìs»�U � � ƨ¸{èE»$Q_è�ûüº[�sI¶a3'4'/>/4�°'70/=�Æ�ëh� 1 � òN ¸{ìs»s(23¨4�4'/GÄE3'/Cò�ëî¸=� 1 � ò � ¸Qìs».» � *_£

�aò)é@��� �¹î ï�ð�ñ � � � � î ï�ð�ñ � û � � õJƨ¸{èE»0Dýj¸{èE».õR�}¸{èE»$Q_è�é�o��Oëh� 1 � ò � ¸Qìs»I5s�'/3�°xÜäû ï ò ¸{òE»�-0���.8:709'=b-08:3 82¥�¸Q®'$0¹±_»�(23'4|=@(2À_-03'«¯��ûLÜ�óÇÆù-¤3 ¸{®'$0¹>±X»j70/n(24¨��=.8 ÜäûÈƨ$ 5s�¨-¤��«:-0�:/>�Åj¸�ï ò »%û � 1 � òN ¸{ìs»>$&¨8:,�á#�~ó©�ùó C °:8q3'/�¢>8:3'�b-¤4¨/,s=.�'/m(24/. 8:-03_=�8q�)/>,@(2=b8:,}ï Nò $�5s�¨-¤�}8:�E/,G(2=.8q,�-0��70-03'/n(+,�¢8q3X=b-03X9¨8:9'�w¥{,.8q¡�¸1�ä1 � ò ¸{ìs»b»1�=.8h�ä1 � ò � ¸{ìs»$Ú�}£"4'/GÄE3'-0=.-08:3�°�¥{8:,m(23_£ w ëü¸1�ä1 � ò ¸{ìs»b»1�a°�Æ�û ï Nò ¸ w »�-0�m=.�¨/�9'3'-0Å_9'/C/>7¤/>¡�/3_=�8+¥<�ä1 � ò � ¸{ìs»��.9¨¢@�=.�E(2=

�aò)é Æ�� �¹î�ï�ð�ñ � � � � î�ï�ð�ñ � û � w é.ï ò ¸{òE»$� �¹î ï�ð�ñ � � � î ï�ð�ñ é"o�ò ëî¸=� 1 � ò � ¸{ìs»b» � Ià -03'¢/©Åj¸�ï ò »�ûè� 1 � òN ¸{ìs»s(+3'4�¥{,.8q¡�=.�¨/�4'/GÄ)3¨-¤=b-08:3�8+¥�ï ò ¸Qò)»>°�Æ�ûåï Nò ¸ w »}-0��(270�.8j=.�'/V9'3'-0Å_9'/��b8:709'=.-08q3�82¥Æ ë\� 1 � ò � ¸{ìs»é� � õ�Ü�¸Qè)»"Dýj¸{èE».õJƨ¸{èE»$Q_è�ûè� w éGÜ�� �¹î�ï�ð�ñ=� � � î�ï�ð�ñ é"o�Ü�ëh� 1 � òN ¸{ìs»�I ¸{®'$0¹-J:»5s�'/,b/G¥{8q,./�¸Q®'$0¹�Jq»��)(+��(j9'3'-0Å_9'/m�.8q7¤9¨=.-08:3�¥{8:,s(23_£ w ë�¸=�ä1 � ò ¸{ìs».»=��(23'4"(+3X£�á���ó���ó C $}�82§�°¨¢8:3¨�.-04'/, w -03O¸1�ä1 � ò ¸Qìs».»=��¥Q8:,x(+707.ó g á'$s5s�¨/3�°E=.�)(+3'À_��=b8�-0=.��9'3'-0Å_9'/3'/�b�n°<=b�'/��b8:709'=.-08q3�8+¥�¸{®'$0¹-J:»}4'8_/�3'8q=s4'/�E/3'4"8q3¤ó_°'(+�]ó��b�)(23¨��¸{á'é@ó-C»$%5s�'/>,./G¥{8q,./:°E=b�'/,./V/Á_-¤�b=.��(j9'3¨-¤Å_9'/m�b8:709'=.-08q3�=.8Æ ë«� 1 � � ¸Qìs»é"op���åá'é� � ýC¸Qè)»bõJÆ'¸{èE»�DnõR�}¸Qè)»�QXè�û � w é*��� �¹î ï�ð�ñ � � � î ï�ð�ñ é"op�ùë\� 1 � òN ¸Qìs»é"opó g áqI¶a3�8:,b4'/,�=b8C¢>8:¡��¨7¤/>=./m=.�'/m�',b8_82¥�82¥�#%,.8q�)8q�.-0=.-08:3�®'$iá'°_-0=}-0�s�.9_â�¢-0/3_=s=b8�=@(2Àq/ w �.9'¢G�"=b�)(2=�°q¥{8:,�(+7¤7�ó g ᨰ

� w é*��� �¹î ï�ð�ñ � � � î ï�ð�ñ û � � ýC¸Qè)»�;�¸{èE».D�õR�}¸{èE»$Q_è�ñ~�Q÷�é*��� �¹î ï�ð�ñ � � � î ï�ð�ñ é"op�ùë\� 1 � ò ¸Qìs»I}�8q=./j=.�)(+=�¸{®'$0¹�®:»s-0�m�@(+=.-0�aÄE/4�¥{8:,�(+7¤7��|ëù�ä1 � ò ¸{ìs»m(23¨4­3'8q=m8:3'70£�¥{8:,�(+7¤7��|ëù� 1 � òN ¸{ìs»��.-03'¢>/j¥{8:,m(�¢>8:3'�b=@(23_=¥{9'3¨¢=.-08:3H�}°<¸{®¨$0¹n®:»½¥{8q7¤7082§s�%¥Q,.8:¡ ÷�¸{ìs»�ûüº'$

jsÎ´Í É:�\�%Øg�.Õ��)��Êr 3'/�3'82§|�.=@(+=./�m(j�.=G(2*'-070-0=a£C,b/�.9'70=�8:3�=.�'/m�b8:709'=.-08q3�82¥�¸Q®'$ ±X»w8:,�¸Q®'$0¹:¹�»�§s-0=.��,./�b�)/>¢=m=.8�ýC°�;Ú(+3'4�÷�$��¬�ç��0ç�{å�æ�å�ç�FsLE°MLË��æ�ª2�|å� �å�æ�Ì­y$ç�¬Ræ�ö�¨z¨[ Z �å��|æ�åQ�Z¨�z�{|ªqæ�å�ç�Fø"¶�µ�¸�ý £ » £�±�² ³ � c P ¸�S�T"¸{ìs»b»@Êm¸Z; £ » £�±�² ³ � ¸�S�?q¸{ìs».» P ¸�ÀfÅ�¸{ò £ » £�±�² ³ � S�?q¸{ìs»�ÒLø"¶-µ�ý ë c P ¸�S�T"¸{ìs»b»é#; ë¸�Sw?:¸Qìs».» P ¸(ÀfÅ�÷�ëz�Ò¸{ìs»�ÒÃDl¾@¾mÇs»Ú¶uµ�·f¸(µZ�

¹:¹

Page 13: Modeling wells in porous media flows - CORE · Modeling wells in porous media ows Pierre Fabrie, Thierry Gallouet To cite this version: Pierre Fabrie, Thierry Gallouet. Modeling wells

Ó�Ò dfehg ºR¾Çsº*·zµ�·f¸(µEý�£sj¢D�jÚa e j¢D�j<Â�´�Ë]¸�ß�ß�j�ë�¶ ÛKP�ÊØÂ�´�Ë�¸�ßÃß\¢ ë"¶ }�¸�ÀfÅ�(qI /�I\´(À�ìYÊ)ý�£�û ¸{ú � £ �3 � 5 »)3 � 5/7 1 �686868 � P¸�ÀfÅ�ýÝû ¸{ú 3 � 5 » 3 � 5�7 1 �686868 � P ¾mÇsº@·Áµ¹·q¸�µ�´(ËH¸�ß�ßqÍ.é)ÎÿëÏ�X¹qé�I-I-I@é@Q2��¸{ú � £ �3 � 5 » £�±�² ³ ¿�¾HÕ@´�ÇsÀfÅ�¶@Åù¿ÃÀ S0T�¸Qìs»h¸�ÀfÅú � £ �3 � 5 ´ ú 3 � 5 (fI /�I ´�À�ì ¸�¾�¢<´ âÌÊ

Ñ�ÒT;�£J´Ð;Í¿�ÀxSw?:¸Qìs» P ¸�¾�¢­´ âÌÊ>�Òmò�£s´�÷ ´�ËRµ�·f¶z×0¶@¸�Ö�ê?µ µ�´@¼f´�ß�´*È(ýz´$Â�¸�V�¸ ìs»b»1��¸�¾®¢m´ â Ð�¿�Ò�¶-Ò � �Cò�£<¸Qè)»���¸Qè)»�Q_èA´ � �u�}¸{èE»$Qq÷�¸{èE»@Ê�¸�¾¢<´ âÌÊf´�ËZ¸�ß�ß��ùë¤V�¸ ìs»�Ô�Ê�¸{ò�£E»*£�±�² ³ü¿�¾KÕ@´�ÇsÀfÅ�¶@Åz¿�ÀxSe1n¸{ìs»Ú¸�ÀqÅ � � ò�£�¸{èE»$Q_è�ûüºK´(Ë�¸�ß�ß�¢�ë�¶ }RÒø"¶�µ%Ü�£«Õ@¶Úµ¹·q¶K¾´�ß�Çsµ�¿�´(ÀHµ�´LÐ)>�Ò�ëmÔx×0¿�µ�·jý�£�Ê!;X£«¸�ÀfÅ�ò�£¯¿�Àؾmµ�¶@¸�Å�´$Â}ýKÊ!;͸(ÀfÅ�ò.ÒÙ�·f¶-ÀsÊ]µ�·f¶-Ë$¶Ú¶*à(¿�¾µÃ¾u¸�¾mÇsÕ�¾¶*Ï-Çs¶�Àqº*¶z´�ÂC¸QÜ £ » £2±�² ³ Ê�¾µ�¿�ßÃß|Å�¶-Àf´(µ�¶@ÅHÕ-ý�¸QÜ £ » £�±�² ³ Ê_¸(ÀfÅHµ�·f¶-Ë$¶Ú¶*à(¿�¾µ�¾Ú¸z¾´�ß¹Çsµ�¿�´�À�Ü~µ�´Ð)>�Ò�>ÔR¾mÇsº@·Hµ�·f¸(µ�Ü £ ´ ܯ¿�À¤�ä1 � � ¸Qìs»@Ê¢¸�¾�¢­´ âÌÊf´(Ë<¸�ß�ß�¹YWù�T�ùQ[�'¸�Q�óå¹�»Òº¹Â�Q�û ásÊ�µ¹·q¶R×.·f´�ß¹¶Ú¾¶@Ï�Çs¶-Àfº@¶�Ü £ º@´(ÀfÉ�¶-Ë�È(¶�¾Xµ�´�Ü�Ê�¿�ÀÛ� 1 � � ¸{ìs»@Ê]¸�¾�¢-´÷âÌÊpÂ�´�ËK¸�ß�ß�¹ÚW ��� áh¸(ÀfÅ�Ü~¿�¾uµ�·f¶ÇsÀf¿�Ï-Çs¶Ú¾m´�ß¹Çsµ�¿�´�ÀHµ�´LÐ)>sÒ�-ÔxÐ$Àq´�µ�¶Rµ¹·q¸�µ�Ð*>sÒ�Ó�Ó*Ôx¿�¾<¿�Å(¶�Àqµ�¿�º@¸�ß|µ�´LÐ)>�Ò�>Ô@Ô�Òº¹ÂKQ�û�®Ý¸�ÀfÅh¿ Â�¸{ò�£E»*£�±�² ³�¿�¾uÕ@´�ÇsÀfÅ(¶*Å«¿ÃÀO¸1�ä1 � � ¸{ìs»b»1��´�ËR¸�ß�ß.� g áØÊ]µ¹·q¶�À µ¹·q¶x×.·q´�ß�¶R¾¶@Ï�Çs¶-Àfº@¶ Ü2£ùº@´�ÀfÉ-¶-Ë�È(¶¾µ�´�Ü�Ê]¿ÃÀ«�ä1 � � ¸{ìs»@Ê�¸�¾©¢u´ âÌÊ.´�ËR¸�ß�ß�¹`W �­� áݸ�ÀfÅ�Üù¿�¾Rµ�·f¶xÇsÀf¿ÃÏ�Çs¶�¾´�ß¹Çsµ�¿�´�À«µ�´\Ð)>�Ò�Ó�Ó�Ô Ð�Àf´(µ�¶z¸�ßB¾´Hµ�·f¸�µ÷�ëî¸=�ä1 � � ¸{ìs»b»1��¸�ÀfÅjò�£<´ª÷z´�ËKµ�·f¶Ú×0¶@¸�Ö�ê?µ\µ�´@¼f´�ß¹´@È(ýR´$Â�¸=�ä1 � � ¸{ìs».»=��ÊØÂ�´�ËK¸�ß�ß�� g á>Ô�Ò5s�'-0�%�.=G(2*'-070-0=a£�,./�b9'70=n°q=.�'/s«q708:*)(+7:¢8q3X�q/,.«q/3'¢/m8+¥)=.�'/s�b/Å_9'/3'¢>/:°'§s-0707_*)/s¢,b9'¢-�(27¨-¤3j=.�¨/s�',.8_82¥<82¥<8:9¨,�¡�(+-¤3m,b/�.9'70=¸��b//�=b�'/��',b8X8+¥�8+¥½5s�¨/8:,b/¡ á¨$0¹�-03 à /¢=.-08q3\J:»>$��¬�ç�ç2y]ç�y��l¬�ç��0ç�{�å�æ-å�ç�FAL�°ML5s�'/j�',.8_8+¥e,b/70-¤/>�x-03O®��b=./�'��$�5s�'/VÄ),.�b=��.=b/�O«:-0�q/��=b�'/�§�/�(2À"¢8q3_�:/,b«:/3'¢>/�-¤3 �ä1 � � ¸{ìs»�¥{8q,x(��.9¨*'�./Å_9'/>3'¢/�8+¥=.�¨/x�b/Å_9'/3'¢>/�¸QÜ�£)»)£�±�² ³�=.82§s(2,b4'�e(j�b8:709'=b-¤8q3�8+¥e¸{®¨$ ±X»°q=.�¨/x�b/¢8q3'4"�.=b/�"£_-0/704'�}=.�'/m�.=b,.8q3'«�¢8:3_�q/,.«q/3'¢/�(23'4�=.�'/=.�¨-¤,b4��.=./>�"=.�'/m¢8q3_�:/,b«:/3'¢>/C8+¥�=.�¨/�§s�'8:70/��./Å_9'/3¨¢/C=b82§e(+,.4'�s=b�'/�9'3¨-¤Å_9'/m�b8:709'=.-08q3�82¥s¸{®¨$¤¹q¹n»>$��æ�¨[�̱�°<t�$K¨sªf­ª��ç�F�æ�¨s¬2þ�¨sFx��¨Jy$ç�¬Rª {�{G�0{¨�z�{�¨sFx��¨�|&¨,.8:¡ /�.=b-0¡�(2=b/"¸{®¨$ µq»°<8q3'/�4'/4¨9'¢/��=b�)(2=�=b�'/��b/Å_9'/3'¢>/î¸QÜ�£E»*£�±�² ³ÿ-0��*E8:9'3¨4'/4�-03 �ä1 � � ¸{ìs»�¥Q8:,C(+3X£�¹zW �Ñ�Q[�'¸�Q}óî¹�»$%5s�'/>3�°_9'��=b8C(V�.9'*'�b/Å_9'/3'¢>/:°E=.�'/��./Å_9'/3¨¢/�¸{Ü2£)»)£�±�² ³O¢8q3_�:/,b«:/��°)(2�:¢&´ âÿ°q§}/n(2À_70£C=b82§e(+,.4'�}�.8q¡�/Ü�-03h�ä1 � � ¸{ìs»w¥{8q,x(+3X£�¹�Wù�#�³Q[�'¸�Q�óå¹�»$s#%(2�.�b-03'«�=.8�=b�'/�70-0¡�-0=n°_(2��¢C´ âü°_-03î¸Q®'$�J:»m¸Þ§s-0=.�"Ü�£<°¨ý�£�°�;�£"(+3'4ò�£�-03'�.=b/n(24�82¥�Ü�°_ýj°�; (23'4"òE»s8:3¨/x8q*'=@(+-03'�e(j�b8:709'=b-¤8q3�Ü�8+¥e¸Q®'$ ±X»>$&¨9',.=b�'/,.¡�8:,b/:°'=b�'/m¢8:¡��)(2¢=b3'/�b��/¡�*E/4'4'-03'«j82¥��ä1 � � ¸{ìs»�-¤3zS � ¸{ìs»}«:-0�q/��Ü £ ´ Ü�-03X=b8xS � ¸Qìs»°)(+��¢­´ âü°q¥{8q,(23_£"¹�Wù�T�ùQ[�'¸�Q�ó ¹�»$%5s�'/>3�°'8q3'¢/�(2«_(2-03�°q9'��=.8�(j�.9¨*'�./Å_9'/>3'¢/:°�Ü £ ´ÒÜ (qI /(I�8q3 ìm$B%�b=.-0¡�(+=./�¸{®qI ²q»s(270�.8�«:-0�:/>�n°_¥Q8:,x(+3_£&=�ë�¶ Û O °E(�*E8:9'3¨4"-03\ì�1n¸Qìs»s82¥}=.�'/��b/Å_9'/3'¢>/"¸�ï � ¸{Ü £ ».» £�±�² ³ §s�¨/,./jï � -0�=.�¨/�¥Q9'3'¢=b-¤8q3�¥{,b8:¡�¶ Ûä=.8�¶ Ûä4'/GÄE3'/4�*_£�ï��'¸�i�»�û ¡�-03�¸�=<éb¡�(2Á_¸aó�=<é@i�».»$ à -03'¢/mï��'¸{Ü2£)»�´ªï��'¸{ÜE»ÿ(qI /�I�8:3�ìm°_§�/=.�¨/3"4'/4¨9'¢/�=b�)(2=sï��)¸QÜ�£'»�´ªï��'¸{ÜE»}§}/n(2À_70£�-03Lìî1n¸{ìs»�(2��¢<´ âÿ$_���=b8�3'82§�°_§�/m�',b82�:/m=.�¨/�¥Q8:707082§s-¤3¨«C(2�b�./,b=.-08:3'�jç¹q$sÜ £ ´ÒÜ�°'(2��¢<´ âÿ°�(fI /�I�-03�ìm°¨§�/n(+À_7¤£�-03Û�ä1 � � ¸{ìs»s(23¨4�-03zS � ¸Qìs»w¥{8:,�(+3_£"¹]Wù��� Q[�'¸�Qmó ¹�»°á¨$}ï��'¸QÜ�£)»Ò´ªï��'¸QÜ)»>°)(+��¢<´ âü°_§}/n(2À_70£�-03¤ìî1n¸{ìs»w¥{8q,�(23_£<=�ë�¶ Û O °®¨$sÜ�-0�s�b8:709'=.-08q3�82¥s¸{®¨$ ±_»$��æ�¨[�~®�°<t@{æ-¬�ç�F�þ-��ç�F�æ�¨�¬2þ�¨sF���¨Rç2y<ª«{�{G�0{¨�z�{�¨sF���¨�|

¶a3 =.�'-0���b=./��°}8q3'/��',.82�q/��=b�)(2=�õ�Ü2£ª´ õ�Ü�-03�¡�/�(2�.9¨,./:°s(2�<¢E´ âü$�5s�'-0��«q-¤�q/�n°w9'� =.8�(��b9'*'�./>ÅX9¨/3'¢/:°õ�Ü2£¶´Lõ�ܪ(fI /�I�-¤3"ìm$}5s�'/3�°)�b-03'¢/�¸k¥{,.8q¡ à =b/��¹�»}=.�'/��b/Å_9'/3'¢>/"¸{Ü2£�»)£�±�² ³ -0��*E8:9¨3'4'/4"-03h�ä1 � � ¸{ìs»}¥Q8:,�(23_£¹YW���� Q[�¨¸�Q}ó­¹�»}¸{(23¨4�Ü�£J´ÒÜ�-03xS � ¸{ìs»�¥{8q,�(23_£�¹YW���� Q[�'¸�Q}ó­¹�».»°+8:3'/s4'/>4'9'¢/�s=b�)(2=}Ü�£J´ÒÜ�°X(+�Ó¢­´ âü°-03h�ä1 � � ¸{ìs»}¥Q8:,x(+3_£­¹ZW³�#�³Q[�'¸�QÉóå¹n»$�5s�'-0���.=b,.8q3'«�¢8q3_�:/,b«:/3'¢>/C-0�x(�¢8:3¨�./Å_9'/3¨¢/�82¥�(j§}/707�À_3'82§s3�,b/�.9'70=§s�'-0¢G�­-0��,b/¢n(+7¤70/4�-03 M·/¡�¡�(�®'$0¹C(³¥{=./>,�=.�'/��¨,.8_82¥s82¥�#%,.8q�)8q�.-0=.-08q3$ ®¨$�5%8�*)/��¨,./¢-0�b/:°%8:3'/j9'�b/�XM·/¡�¡�(�®'$0¹§s-0=.�O¸ZÆ £ » £�±�² ³ ûä¸�Ô 3 ¸QÜ £ »s» £�±�² ³ ¥{8q,�/n(+¢G�¹Í%ë-�X¹qé�I-I�Ibé*Q2�_$M</>=�9'�s3¨82§ �¨,.82�:/�=.�)(+=eõ�Ü £ ´Òõ�Ü�-03�¡�/�(2�.9¨,./:$%&'8q,e(É«:-0�:/3�¥{9'3'¢>=.-08:3<� ¥{,.8q¡ ì =.8j¶ Ûä(23'4�(�,b/n(27E3_9'¡�*E/,�ö2°8:3¨/�4'/3'8q=./����� a ö���=b�'/��b/=©�2è�ë­ì�U���¸{èE»�aåö��X$�M</=�ã g º¨°_8:3'/��E(2��=b8��',.82�q/m=.�)(+=s¡�/n(+�n¸*� n õ�Ü £ óîõ�Ü n aãÕ�³»�´Òºj(2��¢­´ âü$�M</=gÖ g º¨$�&'8q,�=�ë�¶ Û O (23'4­×jëCÅ N O}°_8q3'/m�)(2�� n õ�Ü £ óùõ�Ü n a�ãÕ���¸� n Ü n a�=��Óج� n Ü £ óîÜ n am×[�ÓØL� � � Ù�� £ é

¹ná

Page 14: Modeling wells in porous media flows - CORE · Modeling wells in porous media ows Pierre Fabrie, Thierry Gallouet To cite this version: Pierre Fabrie, Thierry Gallouet. Modeling wells

§s�'/,b/u� � � Ù�� £ û~� n Ü n ��=��Ó�¬� n Ü £ óîÜ n �m×[�Ó�C� n õ�Ü £ ó�õ�Ü n aùãÕ�X$M</>=x9¨��¢G�'8_8:�b/<=�ëO¶ Û O 7�(2,b«:/�/3¨8:9'«q�­-03"8q,.4'/,m=b8��)(��q/�¡�/n(+�¸*� n Ü n aÇ=K�³»YW¸Ö_$ à -03'¢/�Ü�£C´�Ü�-¤3hS 1 ¸Qìs»°¨¥{8q,(23_£�Ä)Á_/4­×Cë¹Å N O =.�'/,b/C/>ÁX-0�b=.�s�.8q¡�/©¢ 1 ¸�4'/>�)/3¨4'-03'«�8q3&×:»}�b9'¢G�"=.�E(2=�¡�/n(2�¸)� n Ü £ ó�Ü n aª×[�2»�WªÖV¥{8q,�¢za¸¢ 1 $5s�'/3�°<-0=x,b/¡�(+-03'��=b8�¢G�'8_8:�b/J×�ë�¶ Û�N Oå�b9'¢G�­=b�)(2=m¡�/n(2�¸�� � � Ù$� £ »�W~Öj¥{8q,©¢O7�(2,b«:/�/3¨8:9'«q��$�¶a3�8:,b4'/,�=b8�*E8:9¨3'4¡�/�(2�¸�� � � Ù�� £ »°_70/=}9'�}=@(2Àq/<��ûåï Ù ¸QÜ £ ó�ï � ¸{ÜE».»w-03­¸{®¨$�J:»s¸Þ§s-¤=b��Ü £ °:ý £ °�; £ (23'4�ò £ -03'�.=b/n(24�82¥�Ü�°:ýj°�; (+3'4�ò)»>$5s�'-0�s£_-0/704'�

e ã ? ¡�/�(2�¸�� � � Ù�� £ »�W � � ý £ ¸Qè)»bõ�¸{Ü £ ¸Qè)»�ó�ï � ¸{Ü�¸{èE».»b»�Dnõjï Ù ¸{Ü £ ¸{èE»�óOï � ¸QÜ�¸Qè)»b».»�QXèW � � ý £ ¸Qè)»<¸�; £ ¸{èE»�óîõjï � ¸QÜ�¸Qè)»b».»�Dnõjï Ù ¸{Ü £ ¸{èE»%ó�ï � ¸{Ü�¸{èE».».»qQXèmñ«× � ò £ ��� ï1�B��� I ¸{®'$0¹�²:»

r 3'/�¢8q3'�.-04'/>,.��3¨82§å=.�'/V=a§�8j=./,b¡��s8q3�=.�¨/�,.-0«:�_=}�)(23¨4��b-04'/m82¥�¸{®¨$¤¹�²:»>$à -03'¢/"¸Qò £ » £2±�² ³ -¤�m*E8:9'3¨4'/4�-03«S}1n¸Qìs»°�=.�'/��b/¢8:3¨4�8:3¨/�«:8_/��=b8 º¨°�(2��×&´�º'°<9¨3'-ã¥{8:,b¡�70£�§s-0=.��,./>�.�E/¢=�=.8C¢%$5s�'/3�=b�'/,./j/Á_-0�.=b�q× 1 g º��b9'¢G�"=.�E(2= ×ZWª× 1ÃÚ × � ò�£ ��� ï � �p� WuÖ e ã ? I ¸{®'$0¹-[:»Â��b�.9'¡�-03'«®×�� ¹q°_=.�'/ÉÄ),.�b=s=./,b¡�-¤3�=.�'/V,.-0«:�_=}�)(23'4��b-¤4¨/x8+¥e¸Q®'$0¹n²q»�-0�s/Å_9)(+7�=.8

� � ýg£�¸{èE»¸�;X£�¸Qè)»½óîõjï��'¸{Ü�¸{èE».»b»0Dnõjï Ù ¸Þï�� O 1 ¸QÜ�£<¸Qè)»b»�óîï��)¸QÜ�¸Qè)»b».»�QXè§s�'-0¢G��¢8:3_�:/>,.«:/>�n°)(+��¢­´ âü°_=.82§s(2,.4¨�

� � ýC¸Qè)»>¸Z;�¸Qè)»�óOõjï � ¸QÜ�¸Qè)»b».»0D�õjï Ù ¸�ï � O 1 ¸QÜ�¸Qè)»b»%ó�ï � ¸{Ü�¸{èE».».»�Q_è.I5s�'-0�sÅ_9)(23_=b-¤=a£�¢�(23"(270�.8j*E/�/Á_�',./>�.�./>4î(+�

��Û �0ÜXÝ Þ[Ý6Ü�� O ÙVß ýj¸{èE»¸Z;�¸{èE»�óîõjï��'¸{Ü�¸{èE».».»�Dnõjï�� O 1 ¸{Ü�¸{èE».»$Q_è�é§s�'-0¢G��«:8_/�}=.8�º�(2�:×�´ ºm�.-03'¢/mýj¸Z;ùó�õjï � ¸{ÜE».»�D.õjï � O 1 ¸{ÜE»wë\S}12¸{ìs»$�5s�'/3�=.�'/>,./m/Á_-0�.=.�:× ? g º�(+3'4<¢ ? ë"¶ }�.9¨¢@��=.�)(+=

×ÚWm× ? (+3'4&¢Ûa�¢ ?qÚ � � ýg£�¸Qè)»>¸Z;X£�¸Qè)»�óOõjï��E¸{Ü�¸{èE».».»�D�õjï Ù ¸{Ü2£�¸{èE»�ó­ï��)¸QÜ�¸Qè)»b».»$Q_è�WmÖ e ã ? I ¸{®'$0¹�µ:»

5s�'/3�°s¢G�'8_8:�.-03'«A×�û ¡�-03<¸�× 1 é�× ? »"¸Q(23'4m×��L¹n»j70/n(24¨��=.8O¡�/�(2�n¸��q� � Ù$� £_»ÛW á�Ö"-㥮¢ ao¢ ? $å5s�'/>,./G¥{8q,./A¢ a¡�(+Á'¸Z¢ 1 éV¢ ? »"¸�,b/¢n(+7¤7�=.�)(+=&¢ 1 -0��«q-¤�q/3 *_£ª×:»j-0¡��'70-¤/>�C¡�/n(2�¸)� n õ�Ü�£�óÿõ�Ü n a ãÕ�³»ÛW ±0Ö_$ä5s�'-0���',b82�:/��=.�'/¢8q3X�q/,.«q/3'¢/O-03ü¡�/n(+�.9',b/"82¥�õ�Ü�£å=b82§e(+,.4'�"õ�Üü(2�­¢à´ âü$�&¨,.8q¡ à =./��¹q°s8:3¨/­4'/>4'9'¢/��=.�E(2=n°�9'�å=b8å(�.9¨*'�./Å_9'/>3'¢/:°�Ü2£J´ÒÜ�°'(+��¢­´ âü°:-03h�ä1 � � ¸{ìs»w¥{8q,e(+3_£­¹YW�����Q[�¨¸�Q�óå¹n»>$%5s�'-0�s¢8:3'¢>7¤9¨4'/� à =./�"á'$��æ�¨[�mL�°<t ��ç�F�æ�¨s¬2þ�¨sF���¨#y$ç�¬Ræ�ö�¨­$Úö|ç� �¨¤{�¨�z�{�¨�F���¨�|�å�'/>3¤QVûÿᨰ2¥{8q,.¡�9'7�(+=.-08:3'�s¸Q®'$ ±X»}(23'4�¸{®¨$¤¹q¹n»�(+,./m-04'/3_=.-0¢n(+7a$�5s�'/>3�°'9'3¨-¤Å_9'/>3'/�.�s8+¥�=b�'/m�.8:709'=b-08:3�8+¥�¸{®¨$¤¹q¹n»w«:-0�:/>�=.�¨/s¢8:3_�:/>,.«:/>3'¢/�8+¥�=b�'/�§s�¨8:70/��b/Å_9'/3'¢>/ ¸QÜ £ » £�±�² ³ =b82§e(+,.4'�%=.�¨/s9'3'-0Å_9'/s�.8q7¤9¨=.-08:3j82¥�¸{®¨$ ±_»$�Û�/¢n(+7¤7¨=.�)(+=n°q*X£ à =./�á'°_=b�'-0�s¢8:3_�q/,.«q/3'¢/j�'8:704'�}-03h�ä1 � � ¸Qìs»}¥{8q,e(+3X£"¹lW���� á¨$¶a3"=b�'/C¢�(2�./RQ�û ®'°E8:3¨/�9'�./>��8:3'¢>/C(2«_(2-03�=.�'/j,./�b9'70=.��8+¥�¬¤¹�®>¯.$x&¨8:707082§s-03'«�¬0¹�®>¯´°¨=.�'/jÅ_9)(23_=b-¤=a£zó C ¸{(+�'�E/n(2,b-¤3¨«�-03#%,b8:�E8:�.-0=b-¤8q3$iá:»�(+�.�b8X¢>-k(+=./4O=b8�ý £ ¢n(+3O*)/j¢G�'8:�b/3�-03'4¨/�)/>3'4'/3_=.70£"8+¥�¢|¸�-03'4'/>/4¯ó C -0�m*)8q9'3'4'/4�¥Q,.8:¡�*E/7082§*_£��.8:¡�/mÅX9E(23_=.-0=a£�8q3'70£�4'/>�)/3¨4'-03'«�8:3 e (23'4"8q3�=.�'/RS0T *E8:9'3'4 ¥{8:,�ý £ »�(23'4"=b�'/�3'8q,.¡�8+¥½=b�'/j8:�E/,@(+=.8:,�ï ò(2�b�.8_¢-�(2=b/4O=.8�ý £ -¤�j(270�.8�*E8:9'3'4¨/4O*_£"�.8q¡�/jÅX9E(23_=.-0=a£"-03'4'/>�)/3¨4'/3_=C8+¥�¢ü¸Þ-¤=m4'/>�)/3¨4'�C8q3'70£"8:3«ó_° e (23'4O8q3=.�¨/LS0T *E8:9'3'4�¥Q8:,mý £ »>$�5s�¨/3�=b�'/�3'8:,b¡ª82¥sï Nò -0�C(270�b8�*E8:9¨3'4'/4�*X£��.8:¡�/�Å_9)(23_=.-0=a£"-03'4¨/�)/>3'4'/3_=j82¥�¢ÿ¸Þ-0=4'/�E/3'4¨�x8q3'70£�8:3Hó_° e (+3'4�8:3�=.�¨/uS0T *)8q9'3'4�8:3�=.�¨/�¢8:¡��)8q3'/3_=}82¥�ý £ »$w5s�'/,b/G¥{8:,b/C=b�'/m�./Å_9'/>3'¢/"¸{Ü £ » £2±�² ³-0��*E8:9'3'4¨/4O-03~�ä1 � � ¸Qìs»�¥{8:,j(23_£�¹�W �¬� á¨$�5s�¨/3O=.�'/�70-¤¡�-0=s82¥�(�¢8:3_�q/,.«q/3_=��.9'*¨�./Å_9'/3¨¢/"82¥�¸{Ü £ » £2±�² ³ -0�-03'4'//>4­(j�.8q7¤9¨=.-08:3�82¥e¸Q®'$0¹:¹�»s(23'4�3'8q=s8:3'70£�(j�b8:709'=.-08q3�82¥�¸Q®'$ ±X»>$%Â��}¥{8q,s=.�'/�¢�(2�./ÚQCûüá'°_=b�'/�9'3'-0Å_9'/3¨/�.�m82¥%=.�'/�.8q709'=.-08:3�82¥x¸Q®'$0¹:¹�»s70/n(24¨�x=b8�=b�'/�¢8:3_�q/,.«q/3'¢/�8+¥�=.�'/j§s�'8q7¤/j�./>ÅX9¨/3'¢/�¸{Ü £ » £�±�² ³ =.82§s(2,b4'��=b�'/�9'3'-0Å_9'/j�.8q7¤9¨=.-08:3=.8�¸{®¨$0¹:¹n»>$�5s�¨-¤�}¢8q3X�q/,.«q/3'¢/m�'8q704'��-03Û�ä1 � � ¸Qìs»w¥{8:,�(+3_£"¹]Wù�T�åá'$%5s�¨-¤�}¢8q¡��'70/=b/��=.�¨/x�¨,.8_82¥%82¥�#%,b8:�E8:�.-0=b-¤8q3®'$i®¨$

¹n®

Page 15: Modeling wells in porous media flows - CORE · Modeling wells in porous media ows Pierre Fabrie, Thierry Gallouet To cite this version: Pierre Fabrie, Thierry Gallouet. Modeling wells

á�¨�©«©«ª-L�°�±ut@S�ámó¯S �Câ ��ç�©A��ª���æ-F�¨f{�{�ãX|Lø"¶�µ%ì Õ@¶R¸xÕ@´�Ë�¶ß�¿�¸�Àz¾¶-µ¢´�Âs¶ Û P ×0¿�µ�·H¸Yû0Àf¿�µ�¶�ø.¶-Õ@¶¾$È(Ç�¶R»Ú¶@¸�¾ÇsË$¶-Òø"¶�µs¹`W ���åßhWüâ ¸�ÀfÅ�¸ZÆ £ » £�±�² ³ Õ@¶x¸HÕ@´�ÇsÀfÅ(¶@Åz¾¶*Ï-Çs¶�Àqº*¶z¿�ÀzS�á<¸Qìs»ÒJD�¾@¾Çs»u¶Rµ�·f¸(µÓÆ £ ´äÆ«¸(Ò�¶-ÒL´(À"ìYÊ]¸�¾¢<´ â³ÒRÙ�·q¶�À<Æ ë¤S á ¸Qìs»K¸(ÀfÅJÆ £ ´ÐÆÛ¿�ÀxS � ¸{ìs»mÊ¢¸�¾�¢�´ âÁÒ��¬�ç�ç2y]ç�y�á�¨s©«©«ª¹LE°�±M</>=7c ë�¶ Û *)/}�.9¨¢@��=b�)(2= � Æ�£ ���2å � �p� W cÊ¥{8:,%(2707�¢ ë�¶ }�$���£j&)(+=.8:9�© ��M</¡�¡�(¨°VÆ�ë¤SEá)¸Qìs»�(+3'4 � Æ ���2å � �p� W cÚ$¶a3O8:,b4'/,�=b8"�',b82�:/�=.�E(2=®Æ�£A´æÆ�-¤3«S � ¸{ìs»>°�70/=©Ö g º'$��}£sB%«q8:,b82��© �m5s�'/8q,./¡�°�=b�'/,b/�/Á_-0�.=b� (�*E8:,./>7¤-�(+3O�./=ýÍ� ìä�b9'¢G�O=.�)(+=m¡�/n(+�¸�ýx»ZWçÖ�(23'4AÆ�£¹´èÆ�9'3¨-T¥Q8:,.¡�70£�8:3OìAé�ýC$j5s�'/3�°�=b�'/,./�/Á_-0�.=.�#¢�C�ë�¶ } �b9'¢G�O=.�)(+=n Æ�£<¸{èE»�óhƨ¸{èE» n WuÖj-T¥�è�ë ì-é}ý (23¨4Ñ¢Ûaª¢�Cq$%5s�'-0�s£_-0/704'�n°q¥{8q,âÛa�¢�C2°

� � n Æ�£�¸Qè)»�óuƨ¸{èE» n � Q_è�WªÖ � ¡�/n(2��¸{ìs»<ñ �[ê n Æ�£<¸Qè)»�óuÆ'¸Qè)» n � Q_è�éWmÖ � ¡�/�(2�¸{ìs»�ñ � Æ £ óAÆ � �� å �B��� ¸�¡�/n(2�¸Þýx».» 1 � @å éWmÖ � ¡�/n(+�¸{ìs»�ñ�á � c � Ö 1V� @å(23¨4��¨,.82�:/>�s=.�)(+=�Æ £ ´ÐÆ�-03LS � ¸{ìs»>°'(2��¢­´ âü$ë ì e������É���s�Èdq��� ��í�� �½�� ���� É�½�îs��� �����jÈ��>�m�

�O/sÄE,.�.=�,./¢�(2707�=b�'/m4'/GÄ)3'-0=b-¤8q3�82¥½(j�b8:709'=b-¤8q3�=b8�=.�'/m,./>«:9'7�(2,b-0¼/4"�',b8:*'70/¡�$ô�¨(ä7F�å�æ�å�çfFAï"°�±èß¹¶-µ¢Ð=Ñ�ÒÃé@Ô�ê*Ð=Ñ�Ò�Ó��-Ôu·f´�ß¹Åx¸(ÀfÅz¸�¾@¾Çs»u¶�÷ù¿�ÀxS�?+¸{ìs»ÒgD̼ظ(¿�Ë�¸{Ü�éQß)»�¿�¾Y¾m´�ß¹Çsµ�¿�´�ÀH´�ÂKµ�·f¶ÚË$¶1È�Ç(ß�¸�Ë¿ð-¶@żpË�´�Õ�ß¹¶-»�¿ Â

Ü�ëHS�?:¸b¸{º¨é.ï�»�U*ì�1¸Qìs».»��zV�¸b¬iº'ébï}¯=U@S�?>¸{ìs».»>é�ºZWåÜ�W ¹qéÜ'ðwë\S�?2¸.¸Qº'ébï�»U�¸�ì�1�¸{ìs».»=��» ¸�±)$0¹�»

Ü�¸$I0éGº:»%ûüÜ�Cm¸�¿�ÀxS ? ¸Qìs».»>é ¸�±)$iáq»

ß ë¤S T ¸b¸{º¨é.ï�»�U*ì 1 ¸{ìs».»>é%æOëHS T ¸.¸Qº'ébï�»U�¸�S ? ¸Qìs».» P »é ¸�±)$i®q»

� ܨð¸$I0é´ê@»é@��� � ï � � � ï ñ � � ô�¸Þæ�¸{è�é´ê@».»'õ�Ü�¸{è�éaê@»EDnõR�}¸{èE»$Q_èó � � æ�¸Qè�é´ê@»�DnõR�}¸Qè)»bò�¸QÜ�¸Qè�é´ê@».»�QXèmñ � �-ñ Ü�¸{è�éaê@»bö2¸{è�éaê@»·óîø+¸{è�éaê@»bú)¸Qè�é´ê@»*ò�÷�¸{èE»$�}¸{èE»$Q_è�ûüº'éo��OëHì�12¸{ìs»é�´�Ë�¸�Òö�Òwêsëî¸Qº'ébï�»é

¸�±)$ ±_»

æ"û óeýj¸{è�éaênéGÜE»<¸�õmß�¸Qè�é´ê@»<óOþ�¸Qè�é´êné@ÜE».» ¸�±)$�Jq»ó � � æ\DõRôs¸Qè)»�Q_è�û � � ôs¸{èE».úE¸{è�é´ê@»a÷�¸{èE»$Q_èVó � � ôs¸Qè)»bö2¸{è�é´ê@»a÷�¸{èE»$Q_è�éopô�ëHì�12¸{ìs»>é0´�ËZ¸�Ò�¶-Òwê�ëO¸{º'ébï�»I ¸�±)$i²q»

ósÎbÏ ôJ�)l�Ô�Ó�Ö�ÑÒ���Ò� Ù�ÔÕ�\õ�ÑsÓ¶a3j8:,.4¨/,%=.8�(23)(+70£X�b/�=.�¨/��',b/�_-08:9'�%�.£_�b=./¡�°2§�/}-03_=.,.8_4'9¨¢/e(�¡�(2�K¦�¥{,b8:¡üS�?q¸.¸Qº'ébï�»U@S�?2¸{ìs»b»�=.8KS�?2¸b¸{º'ébï�»U@S�?2¸Qìs».»¸{(j�b8:709'=b-¤8q3�8+¥�=.�¨/�,./«q9'7�(2,b-¤¼>/4��',.8q*'70/¡�§s-0707�(2�'�E/n(2,�(+��(VÄ)Á_/4��)8q-03X=s8+¥7¦%»$&¨8:, vÜ�ëHS ? ¸b¸{º¨é.ï�»�U*S ? ¸{ìs».»>°'70/=s9'�s4¨/GÄ)3'/CÜ�ûÁ¦}¸ vÜ<»>$�&�-0,b�.=n°¨=.�'/>,./�/Á_-0�.=b��(j9'3'-0Å_9'/m�.8q7¤9¨=.-08:3jß�82¥

¹±

Page 16: Modeling wells in porous media flows - CORE · Modeling wells in porous media ows Pierre Fabrie, Thierry Gallouet To cite this version: Pierre Fabrie, Thierry Gallouet. Modeling wells

ß�ë¤S T ¸.¸{º¨é.ï�»�U*ì 1 ¸{ìs»b»é � � ß�¸{è�éaê@»�Q_èCûüº'é�¥{8:,�(¨$ /:$<êsëO¸{º¨é.ï�»>é� � ýC¸Qè�é´êné vÜ<¸{è�éaê@»b»<¸�õmß�¸Qè�é´ê@»·óîþ�¸Qè�é´êné vÜ)¸Qè�é´ê@».»b»�DnõRôs¸Qè)»�Q_èû � � ôs¸{èE».úE¸{è�é´ê@»a÷�¸{èE»$Q_èVó � � ôs¸Qè)»bö2¸{è�é´ê@»a÷�¸{èE»$Q_è�é1o�ô�ë¤ì 1 ¸Qìs»é}¥{8q,e(¨$ /:$�ê�ë�¸Qº'ébï�»I

¸�±)$�[q»

5s�'/��',b8X8+¥�82¥x=b�'/�/Á_-0�.=./>3'¢/O(23'4�9'3'-0Å_9'/3'/�b��82¥x�b8:709'=b-¤8q3­ß�82¥�¸�±E$�[:»V-¤��(23 -0¡�¡�/4'-�(�=./�¢>8:3'�b/Å_9'/3'¢/�82¥m=.�'/M�(+Á(�´à�-070«:,G(2¡�M</>¡�¡�(¨$5s�'/3�°'=.�¨/,./�/>ÁX-0�b=.�x(V9'3'-0Å_9'/�ÜO¸Q(23'4�§}/x�b/=xÜ�ûÁ¦}¸ vÜE».»��.8:709'=b-08:3�8+¥

Ü ë¤S�?q¸.¸Qº'ébï�»�Ufì�1n¸Qìs».»X�LV�¸.¬iº'ébï}¯$U*S�?>¸{ìs»b»é�ºZWåÜ�W ¹qéܨð�ë\S�?+¸.¸{º¨é.ï�»�Un¸�ì�1n¸Qìs».»1��»>éÜ�¸$D0éGº:»%ûÿÜ�C2é�aÜ'ð¸�I¤é´ê@»é@��� �Yïk� � � Yï ñ � � ô­¸Þæ�¸Qè�é´ê@».»bõ�Ü�¸Qè�é´ê@»�DnõR�}¸Qè)»�QXèó � � æ}¸{è�éaê@»�D�õR��¸Qè)»bò�¸QÜ�¸Qè�é´ê@».»�Q_èxñ � � Ü�¸Qè�é´ê@»$�}¸{èE».ö+¸{è�éaê@»´÷�¸{èE»$Q_èó � � ø2¸Qè�é´ê@»$�}¸Qè)»bú)¸Qè�é´ê@»a÷�¸Qè)»�Q_èCûüº'é=op��ëHì 1 ¸{ìs»éw¥{8q,�('$ /:$�ê}ëî¸Qº'ébï�»é

¸�±)$iµq»

§s�'/,b/æ}¸{è�éaê@»�ûäósýC¸Qè�é´êné vÜ<¸{è�éaê@»b»<¸{õmß�¸{è�éaê@»·ó­þ�¸Qè�é´êné vÜ<¸{è�éaê@»b».»Eéù¥{8:,�(¨$ /q$�¸Qè�é´ê@»wë ìüí�¸{º'ébï�»I ¸�±)$�tq»

5s�'/m/Á_-0�.=b/3'¢/�(23'4�9'3¨-¤Å_9'/>3'/�.��82¥�(j�.8:709'=b-08:3�Ü�8+¥�¸Þ±)$iµ:»w*E/-03'«�¢7�(2�b�.-0¢n(27a°_�b//�¬0¹nº>¯.°_§}/x8q¡�-0=�-0=.�s�¨,.8_82¥@$¶a3�8q,.4'/>,C=b8­�',b82�:/�=.�'/�/Á_-0�.=b/3'¢/�82¥�(�Ä)Á_/4��)8q-¤3_=j=.8«¦}°�70/=j9'��ÄE,.�.=��¨,.82�:/q°%-¤3 à /¢=.-08q3 ±E$iá'°��.8:¡�/H¸�¼p˿ô�Ë¿/�b=.-0¡�(+=./��8q3 Ü�°}æ (+3'4Oß -03'4'/�E/3'4'/>3X=�8+¥ vÜ�$ à 8q¡�/�8+¥x=b�'/¡�§s-0707�*E/",@(+=.�'/>,�9'�b/G¥{9'707s¥Q8:,�8:9¨,�¡�(2-03O,b/�.9'70=¸��¨,.82�:/>4�-03 à /¢=b-¤8q3HJ:»$%5s�¨/3�°'§}/m§s-07¤7E�',b82�:/m-03 à /¢>=.-08:3�±)$i®V=.�'/m¢8q3_=.-03_9'-0=a£�(23'4�=.�'/�¢>8:¡��E(2¢=b3'/�.�s8+¥7¦|¸0¥{,b8:¡S�?q¸.¸Qº'ébï�»U@S�?2¸{ìs»b»}=.8xS�?q¸.¸Qº'ébï�»U@S�?2¸Qìs».».»s(+3'4�§�/m§s-0707)¢>8:3'¢709'4¨/�§s-0=.��=.�'/ à ¢G�)(+9'4'/,�Ä)Á_/4��)8q-03X=s5s�¨/8:,b/¡­$ósÎ´Í k­�4���.Ùà�X�'Ô��M</>=s9'��,b/¢n(2707<=b�)(2=�ò�-0��(2�.�b9'¡�/>4�=.8�*E/uM·-¤�¨�.¢G�'-0=.¼m¢8q3X=b-03X9¨8:9'�s8q3­¬iº'é�¹G¯<§s-0=.�

ò�¸�^q»sûåº ¥{8:, ^ÚW ºò�¸�^q»sûä¹ ¥{8:, ^Úa ¹ ¸�±)$0¹�º:»

ï"°¹®�°Ã± S�T¥¨f{æ-å�©«ªfæ�¨f{&¨8:,sÜ�ûÁ¦�¸ vÜ)»}§s-0=b� vÜ�ëHS�?q¸.¸Qº'ébï�»U@S�?2¸{ìs»b»°¨8:3'/m�',b82�:/�s�'/>,./j=.�)(+=eºÚWåÜ�Wä¹:$��¬�ç��0ç�{å�æ�å�ç�F¹ï"°�±ß¹¶-µ<Ð=ÑsÒ�é@Ô�ê*Ð=Ñ�Ò�Ó��-ÔL·q´�ß¹ÅÛ¸�ÀfÅh¸�¾@¾Çs»u¶j÷ü¿�ÀHS ? ¸Qìs»Òhþ�¶-µ�µ�¿�À�È�ÜOû�¦}¸ vÜ<»R×0¿�µ�· vÜ�ëùS ? ¸.¸Qº'ébï�»U@S ? ¸Qìs»LÐ�×.·q¶�Ë�¶u¦ ¿�¾Å�¶�û0Àf¶@Åz×0¿�µ�·HÐ��fÒ�]kÔ�ê*ÐZ�qÒ�ÞÔ@Ô�Ê¢´(Àf¶<·q¸�¾

ºZWåÜzWÿ¹(I ¸�±)$0¹q¹n»��¬�ç�ç2y]ç�y��l¬�ç��0ç�{�å�æ-å�ç�F-ï.°�±5�(2À_-03'«u��ûäó�ÜÕ��¸$D0é´ê@»�-03�¸�±)$iµq»°_8:3¨/x�E(2�n°�¥{8:,s('$ /:$�êsëO¸{º¨é.ï�»>°ó�� Ü ð ¸$D0é´ê@»éGÜ � ¸$D0é´ê@»$� �]ï1� � � Yï ñ � � ô�¸�æ}¸{è�éaê@»b».õ�Ü � ¸Qè�é´ê@»"Dnõ�Ü � ¸Qè�é´ê@»$Q_èmñ � � ò�¸{Ü�¸{è�éaê@»b»aæ}¸{è�éaê@»"Dnõ�Ü � ¸{è�é´ê@»$Q_èñ � � ¸QÜ � ¸Qè�é´ê@».» ? ö+¸{è�éaê@»´÷�¸{èE»$Q_è�ñ � � ø+¸{è�éaê@»'Ü � ¸{è�éaê@»búE¸{è�éaê@»´÷�¸{èE»$Q_èCû|ºfIà -03'¢/�ò�¸�i�»�û º�¥{8:,Ki�W º"(+3'4��.-03'¢/�úE¸$D0é´ê@»°�ö+¸$D0éaê@»�(23'4O÷ (2,b/�3'8:3'3¨/«X(+=.-0�:/V¥{9'3'¢>=.-08:3'��°�8q3'/�«:/>=.��¸Þ9'�.-03'«"(270�b8¸{á¨$0¹:¹n»b»

¹�J

Page 17: Modeling wells in porous media flows - CORE · Modeling wells in porous media ows Pierre Fabrie, Thierry Gallouet To cite this version: Pierre Fabrie, Thierry Gallouet. Modeling wells

ó<�aÜ ð ¸$D0é´ê@»éGÜ � ¸�D¤é´ê@»$� �Yïk� � � ]ï ñ y � � n õ Ü � ¸{è�éaê@» n ? Q_è�WåºfI¶a3_=./«q,@(+=.-03'«�=.�¨-¤�s7�(+�.=}-¤3¨/Å_9)(270-0=a£�82�:/,�¸Qº'é@in»}(23¨4�9¨�.-03'«

¹á ¸ � Ü � ¸$D0é*i�» � ?� � �B��� ó � Ü �C � ?� � �B��� »%ûäó � íC � ܨð@¸�D0éaê@»>é@Ü � ¸$D0éaê@»�� �Yïk� � � Yï Q+êné

8:3¨/�4'/4¨9'¢/��=.�)(+= � Ü � ¸$D0é@in» � ?� � �B��� W � Ü �C � ?� � �B�p� ¥{8q, (+707¢i�ë ¬iº¨é.ï}¯e(+3'4O=.�¨/,./G¥Q8:,./q°��b-03'¢/"Ü�C¤a º"(¨$ /:$0°%=.�)(+=Ü�¸$D0é*i�»�a ºj('$ /:$0°q¥{8q,�(2707�imëO¬ º¨é.ï}¯.$5�(2À_-03'«u��û ¸QÜjóå¹n» O ¸�D0éaê@»w-03­¸�±E$iµ:»°_8q3'/m�)(2��°_¥{8q,e(¨$ /:$�êsëî¸Qº'ébï�»°

� Ü ð ¸�D0éaê@»>én¸QÜ�óå¹�» O ¸$D0é´ê@»$� �ö]ï1� � � Yï ñ � � ô�¸�æ�¸Qè�é´ê@».»bõ�¸{ÜVóå¹n» O ¸Qè�é´ê@»�Dnõ�¸QÜVó ¹�» O ¸{è�é´ê@»$Q_èó � � ò�¸QÜ�¸Qè�é´ê@».»�æ�¸{è�é´ê@».D�õ�¸{Ü�¸{è�éaê@»·óå¹n» O Q_èñ � � Ü�¸Qè�é´ê@»¸QÜ�¸Qè�é´ê@»·ó ¹n» O ö2¸{è�éaê@»´÷�¸{èE»$Q_èVó � � ø2¸Qè�é´ê@»¸QÜ�¸Qè�é´ê@»<ó ¹n» O úE¸{è�éaê@»´÷�¸{èE»$Q_èCûüºfI

¸�±)$0¹�á:»

à -03'¢/�ò�¸�i�»�û ¹�¥{8:,]ila ¹q°:4'-0�'æ}¸$D0é´ê@»�ûüú)¸�D0éaê@»´÷�óîö2¸�D0éaê@»´÷"8:3�ìÿ(+3'4�æ}¸$D0éaê@»"D@FOûåºj8:3zG)ìm°¨=.�'/�¥{8q7¤7082§s-03'«m�'8q7¤4¨�ó � � ò�¸QÜ�¸Qè�é´ê@».»�æ�¸{è�é´ê@»�Dõ�¸{Ü�¸{è�éaê@»�ó ¹n» O QXèjû ó � � æ�¸{è�é´ê@»�Dõ�¸{Ü�¸{è�éaê@»�ó ¹n» O QXèjû� � 4'-0�_æ�¸Qè�é´ê@»¸QÜ�¸Qè�é´ê@»·ó ¹n» O Q_è�û � � ¸{úE¸{è�éaê@»´÷�¸{èE»�óîö2¸Qè�é´ê@»a÷�¸Qè)»b»¸QÜ�¸Qè�é´ê@»<óå¹�» O Q_è.I

5s�'/3�°�¸�±E$0¹ná:»w*E/¢8:¡�/�

�aÜ ð ¸$D0é´ê@»é�¸{ÜVó ¹�» O ¸�D0éaê@»�� �Yïk� � � ]ï ñ � � ô�¸Þæ�¸{è�é´ê@».».õ�¸QÜVó ¹�» O ¸{è�é´ê@»�Dõ�¸{Üjóå¹n» O ¸{è�éaê@»�QXèjû� � ¸{ø2¸Qè�é´ê@»�óå¹n».úE¸Qè�é´ê@»a÷�¸Qè)»>¸{Ü�¸{è�éaê@»·ó ¹�» O Q_èVó � � ¸{Ü�¸{è�éaê@»�óå¹�».ö2¸Qè�é´ê@»a÷�¸Qè)»>¸{Ü�¸{è�éaê@»Eóå¹�» O Q_è.Ià -03'¢/�ø2¸Qè�é´ê@»�W ¹�(+3'4��b-03'¢/CúE¸$D0é´ê@»°'ö2¸�D0éaê@»}(23'4�÷î(+,./�3¨8:3'3'/>«X(2=b-0�:/s¥Q9'3'¢=b-¤8q3'�n°¨=.�'-0�s£_-0/704'�

� Ü ð ¸$D0éaê@»>én¸QÜVó ¹n» O ¸$D0é´ê@»$� �]ï1� � � Yï ñ � � ô�¸�æ}¸{è�éaê@»b».õ�¸{ÜVó ¹n» O ¸{è�éaê@»�Dõ�¸QÜjó ¹�» O ¸Qè�é´ê@»$Q_è`W ºqI¶a3_=./«q,@(+=.-03'«�=.�¨-¤�s7�(+�.=}-¤3¨/Å_9)(270-0=a£�82�:/,�¸Qº'é@in»}(23¨4�9¨�.-03'«

¹á ¸ � ¸{Üjóå¹�» O ¸$D0é@in» � ?� � �B�p� ó � ¸{Ü C ó ¹�» O � ?� � �B��� »�û � íC �aÜ ð ¸$D0é´ê@»é�¸{ÜVóå¹n» O ¸�D¤é´ê@»$� � ï � � � ï Q+êné

8:3¨/�8:*'=G(2-03'� � ¸{Ü ó ¹�» O ¸�D0é*i�» � ?� � �B��� W � ¸{Ü C óå¹n» O � ?� � �B��� (23¨4"=.�'/>3�°E�.-03'¢/�Ü C W ¹j('$ /:$0°)Ü�¸$D0é@in»�WÚ¹�('$ /:$0°_¥Q8:,x(+707imëO¬iº'ébï}¯.$<5s�'-0�s¢8:¡��'70/=./>�s=.�'/m�',b8X8+¥�82¥½#½,.8:�E8:�b-0=.-08:3�±)$0¹q$ï"°¹®�°�® Sw?:¸b¸{º'ébï�»U@ì�1¸{ìs»b»�¨f{æ-å�©«ªfæ�¨�O/C(+,./�3'82§ä(2*¨7¤/V=.8�/>�.=@(+*'70-0�.�­(�S�?:¸b¸{º¨é.ï�»�U*ì�1¸Qìs».»�*)8q9'3'4"§s�¨-¤¢G��4'/�E/3'4¨��8:3'70£�8:3"=b�'/�3'8q,.¡�82¥�úq÷�(23¨4­öG÷-03LS}12¸.¸Qº'ébï�»�UfS}12¸{ìs»b»$%5s�'-0�}/�.=b-¤¡�(2=b/:°'(2�}=.�'/�(+*)82�q/lS�TZ� *E8:9'3¨4�°_§s-07¤7E*E/x�q/,.£�9'�./ ¥{9'707<-03��',b82�X-03'«V82¥�8q9',s¡�(+-03,./>�.9'70=C¸ à /¢=b-08:3HJ:»>$��¬�ç��0ç�{å�æ�å�ç�F¹ï"°¹®ø"¶�µ]Ð=Ñ�ÒÃé@Ô�ê*Ð=Ñ�Ò�Ó��-ÔX·q´�ß¹ÅÛ¸�ÀfÅH¸�¾@¾mÇs»Ú¶m÷³¿�ÀHSw?:¸Qìs»ÒÚþ�¶-µ�µ�¿�À�È�Ü�û ¦}¸ vÜ)»R×�¿�µ¹· vÜ�ë Sw?:¸b¸{º'ébï�»U@S�?2¸Qìs».»xÐ�×.·q¶�Ë�¶<¦ ¿�¾Å�¶�û0Àf¶@Åz×0¿�µ�·HÐ��fÒ�]kÔ�ê*ÐZ�qÒ�ÞÔ@Ô�Ê¢µ�·f¶-Ë$¶K¶@à(¿�¾µ�¾]VÍÅ�¶=¼Ø¶-ÀfÅ(¿�À�Èz´�ÀØß¹ýx´(À�ò ¾Çsº*·Ûµ¹·q¸�µÃÊ�´�Ë�¸�ß�ß_êsëî¬iº'ébï}¯1Ê

� Ü�¸kê@» � ?� � �B��� ñ�á y � ðC � � n õ Ü�¸Qè�é Y'» n ? Q_è�Q[YÚW � Ü�C � ?� � �B��� ñ�V � ðC ¸ � úE¸�D¤é Y'»_÷ ��� ï1�B��� ñ � ö2¸$D0é Y'»_÷ ��� ï�� �p� »�Q�YsIV¸�±)$0¹�®:»¹n²

Page 18: Modeling wells in porous media flows - CORE · Modeling wells in porous media ows Pierre Fabrie, Thierry Gallouet To cite this version: Pierre Fabrie, Thierry Gallouet. Modeling wells

��¬�ç�ç2y]ç�y��l¬�ç��0ç�{�å�æ-å�ç�F-ï.°¹®5�(2À_-03'«u�ù/Å_9)(27�=.8�Ü�¸$D0éaê@»w(2��(j=./>�.=s¥Q9'3'¢=b-¤8q3�-¤3�¸�±E$ µq»°_8q3'/m8:*'=G(2-03'�n°q¥Q8:,s('$ /:$�êsëî¸Qº'ébï�»°

� ܨð¸$D0é´ê@»éGÜ�¸�D¤é´ê@»$� �Yï�� � � ]ï ñ � � ô�¸Þæ�¸{è�é´ê@».».õ Ü�¸{è�éaê@»"Dõ Ü�¸Qè�é´ê@»$Q_èVó � � ò�¸QÜ�¸Qè�é´ê@».»�æ�¸{è�é´ê@»�Dõ�Ü�¸Qè�é´ê@»$Q_èñ � � ¸{Ü�¸{è�é´ê@».» ? ö2¸{è�éaê@»´÷�¸{èE»$Q_èVó � � ø2¸Qè�é´ê@»'Ü�¸{è�éaê@»bú)¸Qè�é´ê@»a÷�¸Qè)»�Q_èxûüº'é

§s�'-0¢G��«:-0�:/��°_9'�.-03'«�4¨-¤�_æ}¸$D0éaê@»�ûÿúE¸�D¤é´ê@»a÷�óîö+¸$D0éaê@»´÷"8:3"ìü(23'4�æ�¸�D¤é´ê@»�D@F­ûüºj8:3zG)ìm°� ܨð¸$D0é´ê@»éGÜ�¸�D0éaê@»�� �Yï�� � � Yï ñ � � ô�¸�æ}¸{è�éaê@»b».õ Ü�¸Qè�é´ê@»�D�õ Ü�¸{è�é´ê@»$Q_è�ñ � � ' Ü�¸Qè�é´ê@» ? óu;�¸{Ü�¸{è�é´ê@».» ( ö+¸{è�éaê@»´÷�¸{èE»$Q_èó � � ¸aø2¸{è�é´ê@»'Ü�¸Qè�é´ê@»Eóu;�¸QÜ�¸Qè�é´ê@».»b»XúE¸Qè�é´ê@»a÷�¸Qè)»�QXèjûüº'é

§s�'/,b/®;�¸QÜ)»%û � ÞC ò�¸�Y'»$Q[Y<$5s�'/3�°'/�b=.-0¡�(+=./C¸Þ±)$0¹n®q»%¥Q8:707082§s��/n(2�b-07¤£�¥{,.8q¡ (23�-03_=./«q,@(+=.-08:3�82�:/,s=b-¤¡�/s82¥%=.�'/��¨,./�_-08:9'�s-03'/>ÅX9E(270-0=a£"¸�±)$0¹q¹n»$

&¨8:,}�@(2Àq/�82¥%¢8:¡��'70/=./>3'/�.��°'70/=s9'�s«q-0�:/m3'82§å�b8:¡�/�*)8q9'3'4'�s8q3�ß"(23'4�æ�$��¬�ç��0ç�{å�æ�å�ç�F¹ï"°MLø"¶�µ¢Ð=ÑsÒ�é@Ô�ê*Ð=Ñ�Ò�Ó��-ÔK·f´�ß�Åx¸(ÀfÅRß�¶�µ<÷�ë¤Sw?:¸Qìs»ÒYþ�¶-µ�µ�¿�À�È�Ü�û³¦}¸ vÜ)»Z×0¿�µ�· vÜ�ë\S�?q¸.¸Qº'ébï�»U@S�?2¸Qìs».»KÐ�×.·f¶-Ë$¶Y¦³¿�¾<Å�¶�û0Àf¶@Å×0¿�µ�·HÐ��fÒ�]kÔ�ê*ÐZ�qÒ�ÞÔ@Ô�ÒRÙ�·f¶-ÀsÊÓ�ÒzÙ�·f¶-Ë$¶Ú¶*à(¿�¾µ�¾]V 1 Å�¶=¼Ø¶-ÀfÅ(¿�À�Èz´�ÀØß¹ýx´(À\µ�·f¶ÚÀf´(Ë»K¾Ú´$Â�úh¸�ÀqÅ ö�¿�ÀxS�T"¸.¸Qº'ébï�»U@V�¸ ìw».»@Ê � ÷ � � � �B����� Ê r Ê vw ¸�ÀfÅe ¾Çsº*·hµ�·f¸�µ

� õ�ß � ��÷ �Ã� C � H ��ø � � � �B����� � � W�V 1 é � æ � ��÷ �Ã� C � H ��ø � � � �B����� � � W V 1 é ¸�±)$0¹>±X»

Ñ�ÒTR.´(Ë]¸�Àqý]����Q[�'¸�Qwó ¹�»@Ê�µ¹·q¶�Ë�¶<¶@à(¿�¾µ�¾¢V ? Å�¶=¼Ø¶-ÀfÅ�¿ÃÀ�Èx´(Àsß¹ýZ´�Àzµ¹·q¶�Àf´�Ë»Z¾]´$Â}úz¸�ÀfÅ�öZ¿�ÀuS�T"¸.¸Qº'ébï�»U@V�¸ ì}».»mÊ� ÷ ��� ï1�B�p�Ã� Ê r Ê vw Ê%ìYÊE�z¸(ÀfÅ e ¾Çsº*·hµ�·f¸(µ� õ­ß � � ÷ ��� C � H ��ø � � @@� �p���ù�*� W�V ? é � æ � � ÷ ��� C � H ��ø � � @@� �p���ù�*� W�V ? I ¸�±)$0¹-J:»

²K´(µ�¶Úµ�·f¸�µEß«¸�ÀqÅCæÁ¸�Ë�¶]È�¿ÃÉ�¶-ÀHÕ-ýLÐ��fÒ�]kÔ�ê*ÐZ�qÒ�ÞÔ�Ò��¬�ç�ç2y]ç�y��l¬�ç��0ç�{�å�æ-å�ç�F-ï.°ML5s�'/m�',b8X8+¥�82¥e¸Þ±)$0¹±_»%¥Q8:707082§s�%¥{,.8q¡�=.�¨/�¢G�'8:-0¢/KôOû�ß�¸$D0é´ê@»�-03O¸�±E$�[:»$5s�'/m�',b8X8+¥�82¥e¸Þ±)$0¹�Jq»�-0�s«q-¤�q/3�-03"#%,b8:�E8:�.-0=b-¤8q3�®¨$¤¹q°:�b//JB%�.=b-¤¡�(2=b/C¸{®¨$iµ:»$5%8C8q*'=@(+-03�¢8q¡��)(+¢=.3'/>�.���¨,.8:�E/,b=.-0/��-0=},./¡�(2-03'�}=.8�/�b=@(+*'70-¤�b���.8:¡�/�*E8:9'3¨4�8q3"Ü'ðG$��¬�ç��0ç�{å�æ�å�ç�F¹ï"°ïD�¾@¾Çs»u¶zÐ=Ñ�Ò�émÔ>ê*Ð=ÑsÒ�Ó��>Ô¤¸(ÀfÅ�÷|ë S�?q¸{ìs»b»Òzø.¶-µ�ÜOû ¦�¸ vÜ<»R×0¿�µ�· vÜùë S�?q¸.¸Qº'ébï�»U@S�?2¸{ìs»b»xÐ�×.·f¶-Ë$¶Ú¦ ¿�¾uÅ(¶�û0Àq¶*Å«Õ-ýÐ��fÒ�]kÔ�ê*ÐZ�qÒ�ÞÔ@Ô�ÒRÙ�·f¶-ÀsÊÓ�ÒzÙ�·f¶-Ë$¶x¶@à(¿�¾µ�¾uV � Å�¶=¼Ø¶-ÀfÅ�¿ÃÀ�È«´�ÀØß¹ýh´�À"ï�Ê � ú � � ÷ ��� C � H �Qø �7� �p�Ã� Ê � ö � � ÷ ��� C � H ��ø �7� ����� Ê � ÷ ��� � �B�p� Ê r Ê vw Ê�ôÛÊeì¸�ÀfÅ e ¾Çsº@·Hµ�·f¸�µ

� Ü'ð ��� � ��� C � H ��ø �ö]ï��B�p�Ã� � � WùV:�:é ¸�±)$0¹�²:»Ñ�ÒTR.´(Ë]¸�ÀqýK� g QqÊ|µ�·f¶-Ë$¶K¶@à(¿�¾µ�¾�V   Å(¶1¼f¶-ÀfÅ�¿�À(Èx´(Àsß¹ýR´(ÀCï�Ê � ú � ��÷ ��� C � H ��ø ��� ���Ã��Ê � ö � ��÷ ��� C � H �Qø �7� ������Ê � ÷ ��� ï1�B��� Êr Ê vw Ê e Ê�ôÛÊ�ì ¸�ÀfÅR��¾Çsº*·Ûµ¹·q¸�µ

� Ü ð � � � ��� C � H ��ø �¹î�ï�ð @��B����� � � WùV: �I ¸�±)$0¹-[:»¹�[

Page 19: Modeling wells in porous media flows - CORE · Modeling wells in porous media ows Pierre Fabrie, Thierry Gallouet To cite this version: Pierre Fabrie, Thierry Gallouet. Modeling wells

��¬�ç�ç2y]ç�y��l¬�ç��0ç�{�å�æ-å�ç�F-ï.°ï&¨,.8:¡ ¸�±)$iµq»°_8:3¨/x�E(2�}¥{8q,e(+7¤7�ú ë\S�?2¸b¸{º'ébï�»�Ufì�12¸{ìs»b»°ûûû � Ü ð é�ú�� Û � � ��� C � H ��ø�Yï@�B�p�Ã��ß � � Û � � ��� C � H ��ø�]ï@�B������ß ûûû W

V ' � õ Ü ��� � ��� C � H ��ø � � �B����� ñ � æ ��� � ��� C � H ��ø � � �B����� ( � ú ��� � ��� C � H ��ø� ï �B�����ñ�V ' � úq÷ ��� � ��� C � H ��ø � � � �p��� ñ � öG÷ ��� � ��� C � H ��ø � � �B�p�Ã� ( � ú ��� � ��� C � H ��ø� ï �B����� é¥{8q,s�.8:¡�/��b9'-0=@(2*¨7¤/�¢8:3¨�.=@(+3_=<V�$5s�'/3�°'=.�¨/x,b/�.9¨7¤=�¥{8:707082§s�%¥{,b8:¡ ¸Þ±)$0¹±_»�(+3'4O¸�±E$¤¹�®:»>$B%�b=.-0¡�(+=./�¸Þ±)$0¹�[q»<-0�%8:*'=G(2-03'/4j-03C(��.-0¡�-07�(2,E§e(�£_°�9'�.-03'«�¸Þ±)$0¹�Jq»°_¸�±E$¤¹�®:»�(23'4�=b�'/}¥ (+¢=%=.�)(+=¢�ä1 � � ¸{ìs»½-¤��¢8:3_=b-¤3_9'8q9'�.70£-0¡�*)/>4'4'/4�-¤3HS0T�¸Qìs»w¥{8:,]� g QE$B%�b=.-0¡�(+=./�¸Þ±)$0¹n²q»}-¤��9'�./>4O-¤3�=b�'/�¥Q8:707082§s-¤3¨«C�b/¢=.-08q3 ¸ à /¢=b-¤8q3­±E$i®:»�(+3'4"/�b=.-0¡�(+=./�¸Þ±)$0¹�[q»}§s-07¤7�*)/j9'�b/G¥{9'707�¥{8:,�=.�'/�',b8X8+¥�82¥�=b�'/m¡�(+-¤3�,b/�.9¨7¤=j¸ à />¢=.-08:3HJq»$ósÎ�j k�l����4�'Ô��g�<Ô ÑXüÑ�Ë�XÑwÕ�nq���bÑ��¶a3�=b�'-0���b/¢=b-¤8q3�°'§}/sÄE,.�.=s�¨,.82�:/�=.�)(+=�=b�'/�(2�'�¨7¤-0¢n(+=.-08:3R¦O-0�}¢8:3_=b-¤3_9'8q9'�w¥{,.8q¡ Sw?:¸b¸{º'ébï�»U@S�?2¸Qìs».»w=.8j-0=.�./>7T¥G$%5s�'/3�°§}/��.�)(+707<�',.82�q/�=.�)(+=s=.�'/V,@(23¨«:/�8+¥�¦ü¸Þ3)(2¡�/70£¶Åj¸�¦%»sûÄ�>¦}¸ vÜ<»>é vÜ ë\S�?:¸b¸{º¨é.ï�»�U*S�?+¸{ìs».»ý�³»�-¤�s,b/7�(2=b-¤�q/70£�¢8q¡��)(+¢=-03\S�?q¸.¸Qº'ébï�»U@S�?2¸{ìs»b»$m5s�'/>,./G¥{8q,./�=b�'/ à ¢@�E(29'4'/>,mÄ)Á_/4��)8q-¤3_=�=.�'/8q,./¡ §s-07¤7�«:-0�:/V=.�'/j/Á_-0�.=./>3'¢/�82¥sÜ"�b9'¢G�O=.�)(+=Ü�ûÁ¦�¸QÜ)»>$��¬�ç��0ç�{å�æ�å�ç�F¹ï"°Mþø"¶�µ¢Ð=ÑsÒ�é@Ô�ê*Ð=Ñ�Ò�Ó��-ÔK·f´�ß�Å�Ê¢¸(ÀfÅx¸�¾@¾mÇs»Ú¶s÷ù¿�ÀRS�?q¸{ìs»�ÒXÙ�·q¶�Àzµ�·f¶Ú¸@¼(¼�ß¹¿Ãº*¸(µ�¿Ã´�ÀR¦ÁÅ�¶�û0Àf¶@ÅxÕ-ýxÐ��fÒ�]kÔ>ê*Ð��qÒ�ÞÔR¿�¾<¸�º*´(Àfµ�¿�ÀsêÇs´�Ç(¾u»Ú¸@¼(¼p¿�À�ÈYÂË�´�»üS�?q¸.¸Qº'ébï�»U@S�?2¸Qìs».»Kµ�´uSw?:¸b¸{º'ébï�»U@S�?2¸Qìs».»�Ò#%,b8X8+¥�82¥½#½,.8:�E8:�b-0=.-08:3�±)$�JM</>="¸ vÜ £ » £ (+3'4 vÜ��.9¨¢@� =b�)(2= vÜ £ ´ vÜ�-03 S�?+¸.¸{º¨é.ï�»EUfS�?:¸Qìs».»�(+�¶¢E´ âü$¯M</=�¸Tß�é.æ}é@ÜE»m*E/�=.�'/��.8q709'=.-08:3O8+¥¸�±E$�[:»��¸�±E$�t:»s(23¨4�¸Tß £ é.æ £ éGÜ £ »�*)/�=b�'/C�b8:709'=b-¤8q3�82¥x¸Þ±)$�[:»k� ¸Þ±)$�t:»}§s-0=.� vÜ £ -03'�b=./n(+4­8+¥ vÜ�¸��.8�=.�)(+=�Ü £ ûü¦}¸ vÜ £ »m(+3'4Ü�ûÁ¦�¸ vÜ<».»>$ r 3¨/��)(2�s=b8��',.82�q/�=b�)(2=�Ü2£¶´ Ü�-03LS�?q¸.¸Qº'ébï�»U@S�?2¸{ìs»b»$w5s�'/m�',.8_82¥%,./70-0/��8:3�=a§�8j�.=b/�'��$��æ�¨[�̱�° r 3'/mÄ),b�.=s�',b82�:/��=.�)(+=s=.�'/m�b/Å_9'/3'¢/�¸�æ £ » £ ¢8:3_�q/,.«q/��=b8CæO-03LSw?:¸b¸{º'ébï�»�U<¸�S�?:¸Qìs».» P »$5s�'/m4'- A</,./3¨¢/xß2£jó�ß��@(+=.-0�aÄE/�n°�¥{8:,�(+7¤7.ô�ëHS�?q¸.¸Qº'ébï�»U@ì�1¸{ìs».»>°

� HC � � ýC¸{è�é Y_é vÜ2£<¸{è�éVY'»b».õ ñ ß�£<¸{è�éVY¨»�ó�ß�¸Qè�é Y'»)òxDnõRôs¸Qè�é Y'»�QXè�Q[Yñ � HC � � ñ ýj¸{è�éVY_é vÜ�£<¸Qè�é Y'».»�ó�ýC¸Qè�é Y_é vÜ·¸{è�éVY'»b»*òmõmß�¸{è�é Y'».D�õRôs¸{è�é Y'»$Q_è�Q�Yó � HC � �-ñ ýj¸{è�éVY_é vÜ £ ¸Qè�é Y'».»´þ�¸{è�éVY_é vÜ £ ¸{è�éVY'»b»�óOýC¸{è�é Y_é vÜ<¸Qè�é Y'»b»aþ�¸{è�é Y_é vÜE¸{è�é Y'».»)òxDnõRôs¸Qè�é Y'»�QXè�Q[Y�ûüºfI

5s�'/3�°'=@(+À_-¤3¨«C=b�'/�(2��(j=./>�.=s¥Q9'3'¢=b-¤8q3Lô =b�'/m4'- A�/>,./3'¢>/xß £ ó�ß�ûEI £ -¤=w¥{8q7¤7082§s�� õJI £ � ?� � ��� C � H ��ø � � � �B�p�Ã���@� WùV 1 � ¸�ýj¸$D0é-D¤é vÜ £ »�óOýC¸�D¤é-D0é vÜ_»b».õ�ß � ?� � ��� C � H ��ø � � � �B�������@�

ñ<V ? � ýj¸$D0é�D0é vÜ�£:»´þ�¸$D0é-D¤é vÜ2£:»%ó­ýj¸$D0é-D¤é vÜ_»aþ�¸�D¤é-D0é vÜ_» � ?� � ��� C � H �Qø � � � �B�������@� é ¸�±)$0¹�µ:»

¥{8q,s�.8:¡�/�¢8q3_�:/3'-0/3_=�V 1 (+3'4HV ? $ à -¤3¨¢/�=.�¨/��./Å_9'/>3'¢/�¸ vÜ £ » £ ¢8:3_�:/>,.«:/>�x-03zS�?:¸b¸{º¨é.ï�»�UfSw?:¸Qìs».»�¸Þ§s�'-0¢G��¢�(23�*)/�_-0/§�/>4�(2��S�?q¸{ì í�¸{º¨é.ï�»b».»w=.8�Ü�(+3'4��b-¤3¨¢/m=.�'/�¥{9'3'¢=b-08:3'�}ý (+3'4�þ�(2,./�82¥|bs(+,@(2=b�'/8_4'8q,.£�=a£_�E/:°_=.�¨/�,.-0«q�X=}�)(+3'4�.-04'/m8+¥�¸Þ±)$0¹nµq»%«:8_/��=.8�¼/>,.8)$%5s�¨-¤�s�¨,.82�:/>��=.�E(2=sõmß £ ´Òõmß�-03HS�?q¸.¸Qº'ébï�»U�¸�S�?2¸Qìs».» P »s(2��¢<´ âÿ$�}�82§�°_�.-03'¢/

¹nµ

Page 20: Modeling wells in porous media flows - CORE · Modeling wells in porous media ows Pierre Fabrie, Thierry Gallouet To cite this version: Pierre Fabrie, Thierry Gallouet. Modeling wells

æ £ ¸Qè�é´ê@»%ûåýj¸{è�éaêné vÜ £ ¸{è�éaê@»b»<¸{õ�ß £ ¸Qè�é´ê@»�ó­þ�¸Qè�é´êné vÜ £ ¸{è�éaê@»b».»<é¸�æ�£E»*£�¢8:3_�q/,.«q/��=b8CæO-03LSw?:¸b¸{º'ébï�»�U<¸�S�?:¸Qìs».» P »$%5s�'-0��(2¢G�'-0/�q/�s=.�¨/��',.8_8+¥�82¥ à =b/�O¹:$��æ�¨[�~®�° r 3'/�3'82§|�',.82�q/�s3'82§ü=.�'/m¢8q3X�q/,.«q/3'¢/j82¥�Ü £ =.82§s(2,b4'��Ü�$à =b/�ÿ¹j«:-0�q/��=.�¨/�¢8:3_�:/>,.«:/>3'¢/�82¥�¸�æ £ » £ =.82§s(2,b4'�mæå-03 S�?q¸.¸Qº'ébï�»�U<¸�Sw?:¸Qìs».» P »>$<_��.-03'«"¢>7k(+�.�b-¤¢�(27�¢8q¡��)(+¢=.3¨/�.�,./>�.9'70=.��¬¤¹q¹b¯.°'¬0¹nµ>¯.°2-0=�¥{8:707082§s�<¥{,b8:¡ä=b�'/s/�.=b-0¡�(2=b/�s¸�±E$¤¹�®:»%(23¨4�¸Þ±)$0¹n²q»°�=.�)(+=n°q9'�j=.8�(��.9'*'�b/Å_9'/3'¢>/:°'§}/s¢n(23�(2�b�.9'¡�/=.�E(2=

Ü £ ´ ÿÜ�§�/�(2À_70£�-¤3HS�?+¸.¸{º¨é.ï�»EUfì�1³¸Qìs».»>é¸QÜ £ » ð ´ ÿÜ ð §}/n(2À_70£�-03zS�?:¸b¸{º¨é.ï�»�U<¸�ì�12¸{ìs».»=��»éÜ £ ´ ÿÜ�-03LV�¸b¬ º¨é.ï}¯fUE¸�ì 1 ¸Qìs».» � »Ü2£¶´ ÿÜ�-03LS�?q¸.¸Qº'ébï�»�UfS�?q¸{ìs».»>éÜ2£¶´ ÿÜ"(fI /�I_-03"ìåí�¸{º¨é.ï�»�I

5s�'/3�§�/s¢�(23��E(2�.�}=.8V=.�'/�7¤-0¡�-¤=�-03�¸Þ±)$iµ:»s¸Þ§s-0=.��Ü2£�(23'4�æ�£�-03'�b=./n(+4�82¥�Ü�(23'4�æ�».(2�q¢<´ âÝ(+3'4��',.82�q/s=.�)(+=©ÿÜ�-0��.8q709'=.-08:3j82¥�¸Þ±)$iµ:»>$�5s�¨/,./G¥Q8:,./q°_=.�'/�9'3'-0Å_9'/3'/�b�s82¥<=.�'/��.8q7¤9¨=.-08:3�8+¥½¸Þ±)$iµ:»%-0¡��'70-¤/>��=.�E(2=©ÿÜ�ûÿÜ�(23'4�=b�)(2=�=b�'/�§s�¨8:70/�./>ÅX9¨/3'¢/CÜ2£�¢>8:3_�:/,b«:/�s=b82§e(+,.4'�sÜ�-03xS�?:¸b¸{º¨é.ï�»�U*S�?+¸{ìs».»>$%5s�'-0��/>3'4'�s=.�¨/x�¨,.8_82¥�8+¥½#%,b8:�E8:�.-0=b-¤8q3�±)$�J'$¶a=}-¤�w3'82§å�E8:�.�b-0*'70/�=.8V�',.82�q/x#%,b8:�E8:�.-0=b-¤8q3�á¨$¤¹q°2§s�'-0¢G��«q-¤�q/��/Á_-0�.=b/3'¢/V82¥�(m�b8:709'=.-08q3�82¥�=.�¨/x,b/«:9¨7k(+,.-0¼/4��',.8q*'70/¡¸�=b�)(2=m-0�x(+�.�b9'¡�-03'«"¸Qá'$i²:»k� ¸Qá'$0¹±_»s(23'4"÷�-03\S�?q¸{ìs».»b»�-¤3�=.�'/j�./3¨�./�82¥q%�/ Ä)3'-0=.-08:3�±)$0¹:°¨9'�.-03'«�=.�¨/ à ¢G�)(+9'4'/,�Ä)Á_/4�E8:-03_=�5s�'/>8:,./>¡­$��¬�ç�ç2y]ç�y��l¬�ç��0ç�{�å�æ-å�ç�F«®�°�±¶a3�8q,.4'/>,�=.8"�',b82�:/�#%,.8q�)8q�.-0=.-08:3 á'$0¹:°%-0=�-0�j�.9_â�¢>-¤/>3X=j=.8��',b82�:/�=b�'/�/Á_-0�.=./>3'¢/�82¥�(�Ä)Á_/4 �)8q-03X=É¥{8:,j=.�'/�¡�(2�s��'-03'«ù¦ 4'/GÄ)3'/>4ä*X£ ¸�±)$�[q»��¸�±)$iµq»$ #½,.8:�E8:�b-0=.-08:3ü±)$�JO«q-0�:/��=.�¨/O¢8:3_=.-03_9'-0=a£å8+¥`¦ ¸0¥{,b8:¡÷S ? ¸.¸Qº'ébï�»U@S ? ¸Qìs».»�=b8S�?q¸.¸Qº'ébï�»U@S�?2¸{ìs»b»$�M</=TÅC¸�¦%»s*E/�=.�'/V,@(23¨«:/�8+¥7¦}$}5s�'/ à ¢G�)(+9'4'/,�Ä)Á_/4"�E8:-03_=�=b�'/8:,b/¡�«:-0�q/�s=.�'/V/Á_-¤�b=./3¨¢/C8+¥(jÄEÁX/>4"�)8q-03X=�¥{8:,�¦ �',b82�X-04'/>4"=.�)(+=�Åj¸�¦%»�-0��,b/7�(2=.-0�q/70£�¢8q¡��)(+¢=s-03\S�?q¸.¸{º¨é.ï�»�U*Sw?2¸{ìs»b»$�M</=�9'�m�.�'82§ä=b�'-0��7�(2�b=�',b8:�E/,.=a£��'8q7¤4¨�n$5s�'/"/>3'/,.«q£å/�b=.-0¡�(+=./:°s3E(2¡�/>7¤£uB½�.=.-0¡�(2=./O¸Þ±)$0¹n®q»°}«:-0�:/>��=.�E(2=&Åj¸�¦�»�-0��*E8:9¨3'4'/4ü-03�Sw?:¸b¸{º'ébï�»U@ì�1n¸Qìs».»>°s=.�'//�b=.-0¡�(+=./�¸�±E$¤¹�²:»w«:-0�:/>�s=.�)(+=T�³Ü ð é'Ü ë¹ÅC¸�¦%» �C-0�s*E8:9'3¨4'/4�-03¤Sw?:¸b¸{º'ébï�»é�¸�ìî1n¸{ìs»b»1�a$ à -03'¢/uìî1n¸{ìs»�-¤�s¢>8:¡��E(2¢=b/70£/¡�*E/4'4'/>4�-03xS�?q¸{ìs»s(+3'4��b-03'¢/Kì�12¸{ìs»w-¤�s4¨/3'�./m-03LSw?:¸Qìs»°'(V¢7�(2�.�b-0¢n(27�¢8:¡��)(2¢=b3'/�b��,b/�.9'70=�¬0¹:¹ ¯.°)¬0¹nµ>¯�«q-0�:/�}=.�'/,./>7k(+=.-0�:/�¢>8:¡��E(2¢=b3'/�.�V82¥�ÅC¸�¦%»�-03«S�?:¸b¸{º¨é.ï�»�U*S�?+¸{ìs».»�¸{(+3'4O(270�.8"-03«V�¸.¬iº¨é.ï}¯$U�¸�ì�1G¸{ìs».»=��»$�5s�'-0��¢8:3¨¢709'4'/�j=.�'/�',b8X8+¥�82¥½#½,.8:�E8:�b-0=.-08:3�á'$0¹:$

� Ç �����#d �#d �Òí�� �Ê V�� �����w���G�

�O/�3¨82§ÿ�b=.9'4'£�=.�'/V¢n(2�b/�§s�'/,b/�÷"ëz� O ¸{ìs»>°'§s�'-0¢G��¢>8:,.,b/�.�E8:3¨4'�m=.8�5s�'/8q,./¡�á'$0¹q$��¬�ç�ç2y]ç�y<õlö|¨sç�¬�¨s©÷®�°�±��æ�¨[�̱�°<t��<ç�F0{æ-¬�{���æ-å�ç�Fhç�y]æ-ö�¨¤{�¨�z�{�¨�F���¨î¸Þ÷ £ » £2±�² ³ |M</>= ¸ù, £ » £�±�² ³ *)/�(��b/Å_9'/3'¢/�82¥�¡�8:7070-ãÄ)/,.��°_=.�)(+=�-0�T, £ ¸Qè)»sûË¢ P ,)¸�¢<è)»>°'¥Q8:,x(+707�è�ëO¶ ÛKPj(23¨4î(+7¤7G¢OëO¶ }�NE°E§s-¤=b�,�ëHVKT�¸�¶ ÛKP:éb¶ Û O »}�b9'¢G�"=.�)(+=� ,)¸Qè)»�Q_è�û ¹�(+3'4­¸ n è n aä¹ Ú ,)¸Qè)»�ûüº:»�I

M</>=s÷ £ ûä¸�, £ µ<ÿ÷�» n � °'§s�¨/,./¬ÿ÷�-¤�s=b�'/�/Á_=b/3'�.-08q3�*_£�º�8+¥½÷�8:3�¶ ÛKPq$¶a=�-0��/n(2�b£"=.8��',.82�q/C=b�)(2=�°'¥Q8:,x(+707!¢�ë�¶ }�°E÷ £ ëÝSw?:¸Qìs»°E÷ £ a º�(¨$ /:$�(23¨4 � ÷ £ ��� ï$� �p� Wü÷�¸{ìs»>$��O/�¢7�(2-0¡ =.�)(+=÷�£ª´ ÷�°�(2�J¢�´ âü°�¥{8:,j=.�'/�§�/�(2À(�Zµ�=.8:�E8:708q«:£�82¥<V�¸ ìs»�(23'4�=.�)(+=C=b�'/��b/Å_9'/3'¢>/�¸�÷�£)»)£�±�² ³ä-0�j*)8q9'3'4'/>4�-03¸1� 1 � � ¸Qìs».» � ¥{8q,e(+3X£z� g á'$&�-0,.�.=�°'8q3'/m�',.82�q/�s=.�)(+=s÷ £ ´ ÷"¥{8q,s=.�'/m§}/n(2À(��µ�=.8q�)8q7¤8q«:£j82¥¢V�¸ ì�»>$M</>=]�OëHV��2¸{ìs»m¸�=b�)(2=s-0�]�ù¢8:3_=.-03_9'8q9'�}¥{,.8q¡ ì =b8�¶ Û §s-¤=b��(j¢8q¡��)(+¢=}�.9'�'�E8:,b=�-03�ìs»>$%5s�'/3�°

¹�t

Page 21: Modeling wells in porous media flows - CORE · Modeling wells in porous media ows Pierre Fabrie, Thierry Gallouet To cite this version: Pierre Fabrie, Thierry Gallouet. Modeling wells

� � �}¸Qè)»´÷ £ ¸{èE»$Q_è�û � ² � � ÿ�}¸{èE»*, £ µ<ÿ÷�¸{èE»$Q_è�é§s�'/,b/mÿ�}¸{èE»�ûÁ��¸Qè)»}-ã¥�è�ë ìü(23'4Äÿ�}¸{èE»%ûÿºj-ã¥�èA�ë ìm$à -03'¢/q°)(2��¢�´ âü°

� ² � � ÿ��¸Qè)»),0£Tµ�ÿ÷�¸Qè)»�Q_è®´ � ² � � ÿ�}¸{èE»$QXÿ÷�¸Qè)»%û � � �}¸{èE»$Qq÷�¸{èE»é§}/x¢>8:3'¢709'4¨/�=.�)(+=

� � �}¸{èE»a÷ £ ¸{èE»$Q_èJ´ � � �}¸{èE»$Qq÷�¸{èE»é"o��OëHV��2¸{ìs»I ¸�J'$0¹�»r 3'/C(+7¤�b8C�E(2�� ² � � , £ µ<ÿ÷�¸Qè)»�QXè�û|÷�¸{ìs» ¸�J'$iáq»

(23¨4� ² � ��� � ,0£gµ<ÿ÷�¸{èE»$Q_è�W ÷�¸�ý�£E»é ¸�J'$i®q»

§s�'/,b/�ý £ ûÄ�³è�ë"ìmé'4'-0�.=�¸{è�é@ì � »�Wÿ¹��0¢X�C(23¨4�4'-0�.=�¸{è�é@ì � »%û|-¤3�¥/� n èVó ^ n °�^C�ë­ìg�_$à -03'¢/©�G£2±�² ³�ýg£ û'°_8q3'/m�)(2�s÷�¸Þý�£)»�´ ºC(+��¢&´ âÿ$�5s�'/,b/G¥{8q,./�¸�J'$iá:»}(23'4O¸�J'$i®:»w70/n(24�=.8

� � ÷ £ ¸{èE»$Q_èJ´ª÷�¸{ìs»>é

§s�'-0¢G��°'=b8:«q/=.�'/>,�§s-0=.�O¸�J¨$0¹n»°¨«:-0�:/>�s=.�)(+=�÷ £ ´ª÷�¥{8q,s=.�'/V§�/n(+À(�Zµ�=b8:�E8:708:«q£�82¥¢V�¸ ìs»m¸Þ=.�)(+=�-0��¸�J¨$0¹n»}§s-0=.�HV�¸ ìs»-03'�.=b/n(24�8+¥7V��2¸{ìs»b»$}�82§�°_§}/m�',.82�q/m=.�)(+=s=.�'/��b/Å_9'/3'¢>/�¸Þ÷ £ » £�±�² ³ -0�s*E8:9'3'4¨/4�-¤3O¸=�ä1 � � ¸Qìs».»1�·¥{8:,�(+3_£x� g á'$M</>=u��ëä¸{á¨é*â »$ à -03'¢/�÷ ë ¸=� 1 � � ¸{ìs».» � °�8q3'/C�E(2�Cÿ÷|ë ¸=� 1 � � ¸Þ¶ Û P ».» � ¸{(+3'4 � ÿ÷ � ��î�ï�ð @�� ² � � ��� � W � ÷ � �¹î�ï�ð @@� �p��� � »$5s�'/3�°_*X£�(V¢7�(2�b�.-0¢n(+7)¢G�)(+,@(2¢>=./,b-¤�G(2=b-¤8q3�82¥�¸1�ä1 � � ¸Þ¶ Û P ».»=� °_=b�'/,b/x/>ÁX-0�b=.� w ë¤S � � ¸�¶ Û P »s(23¨4��äë�¸�S � � ¸�¶ Û P »b» P °_§s-¤=b��-�)ûÁ���'¸��sóå¹n»>°_�.9'¢G�"=b�)(2=�ÿ÷�û w ñ�4'-0� � (+3'4

� ÿ÷ � ��î�ï�ð @�� ² � � ��� � û � w � � @ � � ² � � � ñ � � � � � @ � � ² � � ���ù� é§s-0=.�

� � � � � @ � � ² � � �Ã��� û § � ² � � n ��¸{èE» n � � Q_è ¨ ï@ �(23¨4

� � � î ï�ð @ � ² � � � û § � ² � � n �}¸{èE» n � QXè ¨ ï@ ñ § � ² � � n õR�}¸{èE» n � QXè ¨ ï@ é=.�¨/�3'8q,.¡ � D � �¹î�ï�ð @�� ² � � ��� � -0�j=.�'/�¢8:,b,./�b�)8q3'4'-03'«�4'9)(27}3'8q,.¡­$�¸�}�8:=b/�=.�E(2= w (+3'4O4'-0�<¸���»�(+,./�3'8:=j3'/¢/>�.�@(+,.-070£�.8q¡�/�3'8:3'3¨/«X(+=.-0�:/�s¡�/n(2�b9',./>�n$i»5s�'/3�°'§s-0=.�z£�£�û�,4£�µ�ÿ÷"û¿,0£gµ w ñ�4'-0�<¸�,0£gµ���»>°

� £ £ � �¹î ï�ð @ � ² � � ����� W � , £ µ w � � @ � � ² � � � ñ � , £ µ�� � � � @ � � ² � � ��� �W � w � � @ � � ² � � � ñ � � � � � @ � � ² � � �����W � ÿ÷ � �¹î ï�ð @ � ² � � �Ã� � I ¸�J'$ ±_»5s�'/m¡�/�(2�.9¨,./m÷ £ -0�s=.�¨/x,b/�.=b,.-0¢=b-¤8q3"=.8�ì|82¥�=b�'/m¡�/n(+�.9',b/K£ £ $��O/��b�)(2707<4'/4¨9'¢/m¥{,b8:¡ ¸�J'$ ±X»w=.�E(2=s=.�¨/��./Å_9'/>3'¢/¸�÷ £ » £�±�² ³ -0�m*)8q9'3'4'/>4O-¤3 ¸1� 1 � � ¸Qìs».» � $�¶a3'4'//4�°%=.�'/,b/�/Á_-0�.=.�j(�70-¤3¨/n(2,m¢8q3X=b-03X9¨8:9'�m¡�(+�'�'-03'«�� ¥{,.8q¡ � 1 � � ¸{ìs»

á:º

Page 22: Modeling wells in porous media flows - CORE · Modeling wells in porous media ows Pierre Fabrie, Thierry Gallouet To cite this version: Pierre Fabrie, Thierry Gallouet. Modeling wells

=.8Ý�ä1 � � ¸�¶ Û P »m�b9'¢G��=.�E(2=��K�|û��ü('$ /:$�8:3�ìm$"&¨9',.=b�'/,b¡�8:,b/:°���¢n(+3O*)/�¢G�'8:�b/3��b9'¢G��=.�E(2=��K� ë³V�¸�¶ Û P »�-ã¥�ùë¤V�¸ ìs»s(23'4��K�haåº�('$ /:$%8q3�¶ Û P -T¥0�«aåºj('$ /:$%8:3"ìm$0M·/=]�OëHV�¸ ìs»��H�ä1 � � ¸Qìs»°_8q3'/m�)(2�� � ��¸Qè)»´÷ £ ¸{èE»$Q_è�W � � n ��¸Qè)» n ÷ £ ¸Qè)»�Q_è�W � ² � � ��¸ n � n »¸{èE»$£ £ ¸Qè)»�QXèW � ��¸ n � n » � î ï�ð @ � ² � � � � £ £ � �¹î ï�ð @ � ² � � ���Ã� WùV � ¸ n � n » � î ï�ð @ �B��� � £ £ � ��î ï�ð @ � ² � � �����W�V � � � î�ï�ð @@�B��� � £ £ � �¹î�ï�ð @@� ² � � �Ã� � é ¸�J'$�Jq»

§s�'/,b/uVä-0�}=.�'/V3'8:,b¡�82¥���¸{(+��(�70-03'/n(+,�¢>8:3_=.-03_9'8:9¨�s¡�(2�¨�'-03'«�¥{,.8q¡¥�ä1 � � ¸Qìs»}=.8z�ä1 � � ¸�¶ ÛZP2».»°_-0=}4'/�E/3'4'�m8q3'70£8:3�ìÿ(+3'4L�_$��O/�9'�./>4 (+70�.8�(�¢7�(+�.�.-0¢n(+7<,./�b9'70=�8+¥ à =G(2¡��E(2¢¢G�'-�(j¬0¹�t�¯<§s�'-0¢G��«:-0�:/� � ¸ n � n » � î�ï�ð @��B��� û � � � î�ï�ð @��B��� $¶a3'/Å_9)(+70-¤=b-0/��¸�J¨$ ±X»�(+3'4"¸�J¨$ Jq»%«:-0�q/s=.�)(+=�=b�'/��b/Å_9'/3'¢/�¸Þ÷ £ » £2±�² ³ -0��*E8:9'3¨4'/4�-03"¸=� 1 � � ¸{ìs».» � $.}�8:=./m(270�.8V=.�)(+=�8:3'/�)(+��÷ £ ´ª÷�°E(2��¢­´ âü°_¥{8q,s=.�'/�§}/n(+À���µ�=.8q�)8q708:«:£�8+¥�¸1�ä1 � � ¸{ìs».»=��¥{8:,�(+3X£z� g á'°'=b�)(23¨ÀX�s=b8�=b�'/��./>�)(2,G(2*'-070-0=a£82¥��ä1 � � ¸Qìs»}¥{8q,<��ë ¸{á¨é*â »s(+3'4"=.�¨/�9'3'-0Å_9'/3'/>�.��8+¥½=b�'/j7¤-0¡�-¤=b/�}82¥�=.�'/j¢8q3X�q/,.«q-03'«��b9'*'�./>ÅX9¨/3'¢/�j82¥�¸Þ÷ £ » £2±�² ³-03­¸=�ä1 � � ¸{ìs».»=�<¥{8q,s=.�'/m§}/n(2À(��µ�=.8q�)8q7¤8q«:£_$<5s�'-0�s¢8q3'¢709'4'/� à =./>�î¹q$��æ�¨[�~®�°<t �J�G��¬�ç�w�å�©«ªfæ�¨R{�ç� �{�æ�å�ç�F0{YªfF�½ ¨f{æ-å�©«ªqæ�¨f{�|&¨8:,q¢"ë�¶ }�NE°_÷�£�-0�}4'/GÄ)3¨/4"-03�=.�'/m�',b/¢/4'-03'«��.=b/��°'70/=s9¨��(270�.8j4'/GÄ)3¨/©Ö/£�¸0ê@»s(23¨4�ú4£�¸$D0é´ê@»%¥Q8:,s('$ /:$�êsëî¸Qº'ébï�»�*_£Ö £ ¸0ê@»�û庨é)ú £ ¸$D0é´ê@»�ûüúE¸$D0é´ê@»é�-ã¥X�-�jú�Qq÷ £ ûÁ�-��öQq÷ £ éÖ £ ¸0ê@»�û � � ¸{ö2¸Qè�é´ê@»·óùúE¸Qè�é´ê@».»´÷�£�¸Qè)»�Q_è� � ÷�£�¸{èE»$Q_è é)ú £ ¸$D0é´ê@»�ûüúE¸$D0é´ê@»)ñ«Ö £ ¸0ê@»>é�-ã¥s¸{ºÚW�» � � ú�Qq÷ £ � � � ö�Q:÷ £ éÖ £ ¸0ê@»�û � � ¸{úE¸{è�é´ê@»�óîö2¸Qè�é´ê@».»´÷�£�¸Qè)»�Q_è� � úE¸{è�éaê@»´÷�£<¸{èE»$Q_è é)ú £ ¸$D0é´ê@»�ûüúE¸$D0é´ê@»¸.¹wóAÖ £ ¸0ê@».»>é}-T¥ � � úsQq÷ £ g � � öQq÷ £ ¸1aåº:»I}�8q=./�=.�)(+=n°�-¤3O=b�'/��./>¢8:3'4 ¢n(+�./:°GÖ�£�¸kê@» g º�(23'4�°%-03�=.�'/�=.�'-0,b4O¢n(2�b/:°�Ö�£�¸0ê@» ë ¸Qº'é�¹G¯.$�5s�¨/3�°%=.�¨/j¥{9'3'¢=b-08:3�ú4£*E/708:3'«q�s=.8xS0T�¸b¸{º'ébï�»U@V�¸ ì�»b»s(23'4"ú4£<¸$D0éaê@»�aåºj8:3 ì ¥{8q,�('$ /:$·êsëî¸Qº'ébï�»$�&¨9',.=b�'/,.¡�8:,b/:°_8:3'/m�)(+�

� � ú0£�¸{è�éaê@»´÷�£�¸Qè)»�Q_è�û � � ö2¸Qè�é´ê@»a÷�£<¸Qè)»�QXè�é�¥{8q,�('$ /:$�ê}ëî¸Qº'ébï�»é(23¨4�°E�.-03'¢/�ö}ó�ú�ë«S0T"¸b¸{º'ébï�»U@V�¸ ì�»b»°E=.�'/j�./>ÅX9¨/3'¢/"¸ùÖ £ » £�±�² ³ -¤��*)8q9'3'4'/>4­-03\S0T�¸b¸{º¨é.ï�»b»�(23¨4�¸0¥{,b8:¡Ò¸Qá'$0¹±_».»Ö�£¶´LºC(¨$ /:$%8:3�¸{º'ébï�»s(2��¢�´ âü$&¨8:,�¢îë­¶ } N °)70/=j¸{Ü £ ébæ £ éQß £ »�*)/j=.�'/j�.8q709'=.-08:3�8+¥½=b�'/�(2�'�',b82Á_-¤¡�(2=b/4��',b8:*'70/¡Ò¸Qá'$iá:áq»��¸{á¨$ á([:»w§s-0=.�"÷ £ -03'�.=b/n(2482¥s÷å(23'4Oú4£­-03'�b=./n(+4O82¥�ú�¸Þ/Á_-0�.=./>3'¢/�8+¥e�b9'¢G� (��.8q709'=.-08:3�-0�m«:-0�:/3O*_£­#½,.8:�E8:�b-0=.-08:3Oá'$0¹n»>$�5s�'/3�°�¸QÜ�£<ébæ�£�é{ß2£'»�@(+=.-0�aÄE/�nç

Ü £ ëHS�?q¸.¸Qº'ébï�»U@ì�1n¸{ìs»b»��zV�¸.¬iº'ébï}¯$U*Sw?n¸Qìs».»>é�ºKWåÜ £ ¸$D0é´ê@»�Wä¹:é¸QÜ £ » ð ëHS�?q¸.¸{º¨é.ï�»�Un¸�ì�1n¸Qìs».»1��»>é ¸�J'$i²q»

Ü £ ¸$I0éGº:»%ûåÜ C -03HS ? ¸Qìs»é ¸�J'$�[q»ß £ ë¤S T ¸b¸{º¨é.ï�»�U*ì 1 ¸{ìs».»>é%æ £ ë¤S T ¸.¸{º¨é.ï�»�Un¸�S ? ¸{ìs».» P »>é ¸�J'$iµq»

�@¸QÜ�£E»aðG¸$I0éaê@»>é*��� �]ï1� � � Yï ñ � � ô�¸�æ�£�¸Qè�é´ê@».»bõ�Ü�£E¸{è�éaê@»�D�õR�}¸{èE»$Q_èó � � æ £ ¸Qè�é´ê@».D�õR�}¸{èE».ò�¸{Ü £ ¸{è�éaê@»b»$Q_èmñ � � Ü £ ¸Qè�é´ê@»$�}¸Qè)»bö2¸{è�éaê@»´÷ £ ¸Qè)»�QXèó � � ø2¸Qè�é´ê@»$�}¸{èE».ú4£<¸{è�é´ê@»a÷�£<¸{èE»$Q_è�ûüº'éop�ùë¤ìî1n¸{ìs»>é}¥{8:,s('$ /:$�êsëO¸{º¨é.ï�»>é

¸�J'$�tq»

æ £ ¸{è�éaê@»%û óeýj¸{è�éaênéGÜ £ ¸Qè�é´ê@».»K�nõmß £ ¸{è�éaê@»�ó�þ�¸{è�éaênéGÜ £ ¸{è�é´ê@».» ��éù¥{8:,�(¨$ /:$�¸Qè�é´ê@»wë ìüí�¸{º¨é.ï�»>é ¸�J'$0¹�º:»ó � � æ £ ¸Qè�é´ê@»"DnõRôs¸Qè)»�QXè�û � � ô�¸{èE».ú £ ¸{è�é´ê@»a÷ £ ¸{èE»$Q_èVó � � ôs¸Qè)»bö2¸{è�éaê@»´÷ £ ¸{èE»$Q_è�éopô ë¤ì�1�¸{ìs»ée¥{8:,s('$ /:$�êsëO¸{º'ébï�»I ¸�J'$0¹q¹n»

á'¹

Page 23: Modeling wells in porous media flows - CORE · Modeling wells in porous media ows Pierre Fabrie, Thierry Gallouet To cite this version: Pierre Fabrie, Thierry Gallouet. Modeling wells

B%�b=.-0¡�(+=./­¸Þ±)$0¹n®q»m«:-0�q/� (�*E8:9'3'4 -03 Sw?2¸.¸Qº'ébï�»U@ì�1n¸{ìs»b»V¥{8:,j=.�¨/��b/Å_9'/3'¢>/�¸{Ü £ » £ (23'4mB%�.=b-0¡�(2=b/­¸Þ±)$0¹�[q»m«:-0�:/>�(�*E8:9'3¨4O-¤3«S�?q¸.¸Qº'ébï�»U�¸1�ä1 � � »1��»�¸0¥Q8:,�(23_£«� g QX»x¥{8:,m=.�¨/��./>ÅX9¨/3'¢/O¸.¸QÜ £ » ð » £ $�5s�'/3�°%¢>7k(+�.�b-¤¢�(27�¢8q¡��)(+¢=.3¨/�.�¬0¹:¹b¯.°�¬0¹nµ>¯�,b/�.9'70=�70/n(24¨��=b8�=b�'/­,b/7�(2=b-¤�q/"¢8q¡��)(+¢=.3'/>�.��82¥C=b�'/��./Å_9'/3¨¢/ÿ¸QÜ £ » £ -¤3ÁS ? ¸b¸{º¨é.ï�»�U*S ? ¸{ìs».»�(+3'4å-03V�¸b¬ º¨é.ï}¯$U�¸1�ä1 � � »1��»m¸k¥{8:,�(+3X£x� g Q_»$�5s�'/3�°¨9'��=.8�(j�.9¨*'�./Å_9'/>3'¢/:°<8q3'/m¢n(23"(+�.�.9¨¡�/:°'(+��¢<´ âü°

Ü £ ¸�D¤é´ê@»G´ÒÜ�¸$D0éaê@»w-03­¸=�ä1 � � »1�aé(op� g QEé�9'3'-ã¥{8q,.¡�70£�§s-0=b��,./>�.�E/¢=�=b8�êsëO¬ º¨é.ï}¯.éÜ £ ´ÒÜ�§}/n(2À_70£�-03LSw?:¸b¸{º'ébï�»U@ì�1n¸Qìs».»>é¸{Ü £ » ð ´ÒÜ ð §�/�(2À_70£�-¤3HS�?+¸.¸{º¨é.ï�»�Un¸=�ä1 � � »=��»éso�� g Q�I ¸�J'$0¹�á:»

}�8q=./j=.�)(+=xÜ�ë¯V�¸b¬iº'ébï}¯=U�¸1�ä1 � � »=�k»V¸0¥{8q,x(+3X£H� g Q_»°<Ü�ë¯S�?q¸.¸Qº'ébï�»U@ì�1¸{ìs»b»x(+3'4­Ü ð ë¯S�?q¸.¸{º¨é.ï�»�Un¸=�ä1 � � »=��»}¥{8q,(23_£x� g Q)$&¨9',.=b�'/,.¡�8:,b/:°��b-¤3¨¢/LS�?q¸.¸Qº'ébï�»U@S�?2¸Qìs».»j¢n(+3�*E/�-04'/>3X=b-ã¥{£­=b8 Sw?:¸Qì í ¸{º¨é.ï�»b»°%8q3'/�¢n(23 (270�.8­(+�.�b9'¡�/�¸�9'��=.8O(�.9¨*'�./Å_9'/>3'¢/n»j=b�)(2=mÜ�£C´�Ü­(¨$ /�8:3Oìüí�¸{º¨é.ï�»V¸�§s�'-0¢G��-¤�m/Å_9'-0�:(+7¤/>3X=�=.8�Ü�£�¸�D0éaê@»q´ Ü�¸�D¤é´ê@»�(¨$ /:$m8:3­ì ¥{8:,m('$ /:$ê�ë�¸Qº'ébï�».»$:_��.-03'«�=.�¨/C*E8:9'3¨4�8q3­Ü2£�«q-0�:/3�-03O¸�J'$i²q»m¸�3)(+¡�/70£�ºÚWÿÜ2£zW ¹C(¨$ /q$�8:3�ìåíO¸{º¨é.ï�»b»°¨=.�'-0��«q-0�:/��°'-03�)(+,.=.-0¢9¨7k(+,n°'ºZWåÜzWÿ¹�(¨$ /q$%8:3�ìüí�¸{º¨é.ï�»>$5s�'/w¥{9'3'¢>=.-08:3CÜÉ�@(2=b-0�aÄ)/��¸{á'$0¹-[:»°q¸{á¨$¤¹�µ:»<(+3'4�=.�¨/�Ä),b�.=%�)(+,.=�8+¥�¸{á¨$0¹n²:»>$<¶a3'4'/>/4�°29'�b-03'«C¸k¥{8:,�Q�ûÿ®q»<¡�8:,./%�',b/¢-0�./s/>�1�=.-0¡�(2=./>��8:3V*)8q=.�j�./Å_9'/3¨¢/�m¸�÷ £ » £�±�² ³ (+3'4�¸�æ £ » £�±�² ³ ¸Þ3)(2¡�/70£X°>=.�'/}¥a(2¢=�=.�)(+=�=b�'/}�./Å_9'/3¨¢/C¸Þ÷ £ » £2±�² ³ -0�%*)8q9'3'4'/4-03O¸1�ä1 � � ¸{ìs»b»1�·¥{8:,m(23_£L� g á�(23¨4"=.�'/>,./G¥{8q,./j=.�'/V�./Å_9'/3¨¢/"¸�æ £ » £�±�² ³ -0�s*E8:9'3'4¨/4"-03HS0T"¸b¸{º'ébï�»U�¸�S � ¸{ìs»b» P »}¥{8q,(23_£¯�h� áq»°%=.�¨/��b/Å_9'/3'¢>/ ¸b¸{Ü £ » ð » £2±�² ³ -0�j*)8q9'3'4'/4 -03 S ? ¸b¸{º'ébï�»U�¸1� 1 � � ¸{ìs»b» � »É¥{8:,�(23_£¯� g á�§s�'-0¢@��«:-0�:/>�ܨð�ë Sw?:¸b¸{º'ébï�»U�¸1�ä1 � � ¸{ìs»b»1��»°�¥{8:,m(23_£\� g á'°�(+3'4�=.�'/,b/G¥{8q,./�ÜùëÝV�¸.¬iº'ébï}¯$Un¸=�ä1 � � ¸{ìs»b»1��»�¥{8:,m(23_£\� g á'$V5s�'-0��',b82�:/�s=b�'/C(2�b�./,b=.-08:3�8+¥wÛ�/¡�(+,.À�á'$iá'$¶a=s,./>¡�(2-03'�}=.8��¨,.82�:/�¸Qá'$0¹�tq»��¸{á'$iᨹn»%(23'4�=b�'/��./>¢8:3'4��)(2,b=s82¥e¸Qá'$0¹n²q»$�5s�'-0�s§s-07¤7E*E/�4'8:3¨/x-03�=.�'/�3¨/Á_=s�.=./>��$��æ�¨[�mL�°<t �Yª�{�{�å�F�þzæ-ç\æ-ö�¨­ �å�©«å�æ�å�F~tVþE°��Õ|���tVþE°�±�±�|©|�O/sÄE,.�.=s�¨,.82�:/m=b�'/m¢8:3_�:/>,.«:/>3'¢/�8+¥�=.�¨/x§s�¨8:70/��b/Å_9'/3'¢/>� ¸ãß £ » £ (+3'4­¸Þæ £ » £ (23'4�§�/m�',b82�:/C¸Qá'$0¹�tq»°E¸{á¨$ ᨹn»>$�5s�'/�./>ÅX9¨/3'¢/�¸Tß £ » £ �:/,b-TÄE/�n°�¥{8:,�(+7¤¡�8:�b=�/�:/>,.£�ê�ëO¸{º¨é.ï�»>°

� � ýj¸{è�éaênéGÜ�£E¸{è�éaê@»b» ñ õmß�£<¸Qè�é´ê@»<óOþ�¸Qè�é´êné@Ü2£)¸Qè�é´ê@».» ò DnõRô�¸{èE»fQ_èû � � ¸{ú £ ¸Qè�é´ê@»·óùö+¸{è�éaê@»b»$ôs¸Qè)»´÷ £ ¸Qè)»�Q_è�é"opô�ëHì 1 ¸{ìs»�I

¶a3m8:,b4'/,%=.8s�',b82�:/%=.�¨/�¢8:3_�q/,.«q/3'¢/s8+¥X=.�¨/��b/Å_9'/3'¢/>�e¸ãß £ » £ (23'4�¸Þæ £ » £ °�§}/��.�)(+707:(2�'�'70£�=.�'/�,b/�.9¨7¤=�82¥)#%,b8:�E8:�.-0=b-¤8q3®'$i®¨$&�-0,.�.=�°'§}/m,./¡�(2,.À�=b�)(2=}¥Q8:,s(270¡�8q�.=�/�q/,.£�ês-03­¸{º¨é.ï�»>°_8:3'/m�)(+�n°E(2��¢<´ âÿ°

ýC¸�D0éaênéGÜ £ ¸$D0é´ê@».»G´ªýC¸�D0éaênéGÜ�¸�D0éaê@»b»O('$ /:$�-¤3"ìméþ�¸$D0é´êné@Ü £ ¸�D0éaê@»b»!´ þ�¸$D0éaênéGÜ�¸$D0éaê@»b»"('$ /:$%-03"ì�I

}�/>ÁX=�°<�.-03'¢>/�÷�£C´L÷�¥{8:,s=b�'/C§}/n(+À���µ�=b8:�E8:708:«q£�8+¥�V�¸ ìs»�(+3'4CÖ�£�¸0ê@»q´�º ¥{8:,�(¨$ /:$sê�ë ¸{º¨é.ï�»>°E8:3'/C(+70�.8��E(2�n°<(+�¢<´ âÿ°q¥Q8:,�('$ /:$�ê}ëî¸Qº'ébï�»°¸{ú £ ¸�D¤é´ê@»·óùö+¸$D0éaê@»b»a÷ £ ´ ¸{úE¸$D0é´ê@»�óîö2¸�D¤é´ê@».»´÷�¥{8q,s=.�'/m§}/n(2À(��µ�=.8:�E8:708q«:£j82¥7V�¸ ìs»I

¶a3�8:,.4¨/,�=.8�(2�'�¨7¤£�=.�'/j,./�b9'70=�82¥s#%,.8q�)8q�.-0=.-08:3O®'$i®¨°)-0=�,./¡�(2-03'��=b8��.�'82§ =.�)(+=x=b�'/j�./Å_9'/3¨¢/­¸b¸{ú £ ¸$D0é´ê@»Gó�ö2¸�D¤é´ê@».»k�÷ £ » £�±�² ³ -0��*E8:9'3¨4'/4�-03"¸=�ä1 � � ¸{ìs».»=�E¥{8:,}(23_£X� g áV(23'4�('$ /:$·ê�ë�¸{º¨é.ï�»�¸�=.�¨/3�°_8:3¨/�(270�.8��E(2�m¸{úE¸$D0é´ê@»'ó�ö2¸$D0é´ê@».»a÷�ë¸1� 1 � � ¸Qìs».» � (23'4å¸Qú £ ¸�D0éaê@»wó ö2¸�D0éaê@»b»a÷ £ ´Ì¸{úE¸�D¤é´ê@»%óåö2¸�D¤é´ê@».»´÷�¥{8:,m=b�'/�§}/n(2À(��µ�=b8:�E8:708:«q£"82¥�¸1� 1 � � ¸Qìs».» � ¥{8:,�(23_£� g á�(23'4­(¨$ /:$wê�ë ¸Qº'ébï�».»$�5s�'-0�s-¤�m(�¢8q3'�./>ÅX9¨/3'¢/�8+¥½=b�'/V¥ (2¢=s=b�)(2=j¸�÷�£)»)£�±�² ³|-¤��(��./Å_9'/>3'¢/�82¥�3'8q3'3'/«_(2=b-¤�q/¡�/�(2�.9¨,./�s8q3 ìm°_*E8:9¨3'4'/4"-03�¸1�ä1 � � ¸{ìs»b»1�·¥{8:,s(23_£L� g á¨$%¶a3'4'//>4�°'70/=]� g áC(+3'4H�Oëh�ä1 � � ¸{ìs»>°_8:3'/m�)(+�

� � �}¸{èE»¸Qú £ ¸{è�éaê@»�óOö2¸{è�é´ê@».»a÷ £ ¸Qè)»�QXè�W ¡�(+Á[� n ú £ ¸{è�éaê@»�óOö2¸{è�éaê@» n éXè ë ìg� � � n �}¸{èE» n ÷ £ ¸{èE»$Q_èW ¡�(2Á[� n ú4£<¸{è�é´ê@»�óîö2¸Qè�é´ê@» n é_è�ë­ìg� � ¸ n � n » � î ï�ð @ �B�p� � ÷�£ � �¹î ï�ð @ �B�p�Ã� � I ¸�J'$0¹�®:»&¨,.8:¡Ò(�§}/707}ÀX3¨82§s3O,./�b9'70=C8+¥ à =@(+¡��)(+¢¢G�'-�(­¸Þ�.//O¬0¹�t>¯{»>° � ¸ n � n » � î�ï�ð @��B�p� û � � � î�ï�ð @m�B��� $"5s�'/3�°%�.-03'¢/"úE°�ö�ëS0T�¸b¸{º'ébï�»U@V�¸ ì�»b»�(+3'4Ú¸�Ö £ » £2±�² ³ -0��*E8:9'3¨4'/4ä-03³S0T�¸b¸{º'ébï�».»>°C¸�J'$0¹n®q»�«q-¤�q/��=b�)(2=�=b�'/­�b/Å_9'/3'¢>/ÿ¸.¸Qú £ ¸$D0é´ê@»}óö2¸�D¤é´ê@».»´÷ £ » £�±�² ³ -0�s*)8q9'3'4'/>4"-03­¸1�ä1 � � ¸Qìs».»=�<¥{8q,e(+3X£z� g á�(23'4�('$ /:$�êsëO¸{º'ébï�»$

á:á

Page 24: Modeling wells in porous media flows - CORE · Modeling wells in porous media ows Pierre Fabrie, Thierry Gallouet To cite this version: Pierre Fabrie, Thierry Gallouet. Modeling wells

5s�'/3�°'§}/x¢�(23­(+�'�'70-0/4�=b�'/��.=G(2*'-070-0=a£�,./�b9'70=�«q-¤�q/3�-03"#%,b8:�E8:�.-0=b-¤8q3"®'$i®�=b8�¢8:3¨¢709'4'/�=b�)(2=s=b�'/,./V/Á_-¤�b=.�m(j9'3'-0Å_9'/ß�-03LS0T�¸.¸Qº'ébï�»�U��ä1 � � ¸Qìs».»m¸k¥{8:,�(+3X£z�T�åá:»}�b9'¢G��=b�)(2=�°)(+��¢<´ â ¸�§s-0=.�'8q9'=}/Á_=.,@(+¢=.-08q3"82¥%�.9'*'�b/Å_9'/3'¢>/�@»>°

ß�£<¸�D¤é´ê@»G´�ß�¸�D0éaê@»�-03h� 1 � � ¸{ìs»>é"op�T�åá'é�¥{8:,�(¨$ /:$<êsëO¸{º¨é.ï�»�I ¸�J'$0¹>±X»5s�'/3�°)4¨/GÄ)3'-03'«�æ�§s-0=.�"=b�'/mÄ),b�.=m70-03'/m82¥x¸Qá'$iá'¹�»°_=.�¨/�¥{9¨3'¢=.-08q3'��æ (23'4�ß"�G(2=b-¤�´¥{£O¸{á¨$¤¹-t:»s(+3'4�¸{á¨$iá'¹n»>$�}�8:=b/C(270�b8=.�E(2=m=.�'/j�./>ÅX9¨/3'¢/�"¸Qõmß�£<¸$D0é´ê@».»)£�±�² ³ÿ(+3'4 ¸�æ�£<¸$D0é´ê@».»*£2±�² ³ÿ(+,./�*E8:9'3'4¨/4­-03�¸�S � ¸{ìs».» P ¥{8q,x(+3X£h��� ᨰE9'3'-ã¥{8:,b¡�70£§s-0=.��,./>�.�E/¢=m=.8Vêsëî¸Qº'ébï�»°_/Á_¢/�¨=�8:3"(j3'/>7¤-0«q/n(2*'70/��./=�$¶a=m,./¡�(2-03'��=.8��¨,.82�:/�¸{á¨$iá:º:»�(23'4�=.�'/��b/¢8:3¨4O�)(2,b=�82¥�¸{á¨$0¹n²:»>$�_��.-03'«O¸�J¨$0¹±X»�(23'4�=.�'/�4'/ Ä)3'-0=.-08:3�82¥�æ}°<8:3'/j�)(+�æ�£�¸�D¤é´ê@»q´ æ�¸�D¤é´ê@»�-03�¸�S � ¸{ìs»b» P °E¥Q8:,�(27070��� á'$j5s�'/>,./G¥{8q,./:°%ô�¸�æ�£�¸�D0éaê@»b»�´ ô�¸Þæ�¸$D0é´ê@».»�-03\c P ¸�S � ¸Qìs».»°�¥{8:,�(23_£�-��â (+3'4�¥{8:,�('$ /:$ ê�ë ¸Qº'ébï�»$�5s�)(23'À_�j=.8�=.�'/�('$ /:$�¢8:3_�:/>,.«:/>3'¢/"8+¥xÜ�£O=b82§e(+,.4'��Ü�(+3'4�=b�'/LS T � *E8:9¨3'48:3å¸QÜ £ » £�±�² ³ °%§}/�(+70�.8"�)(��q/�æ £ ¸$D0é´ê@».ò�¸{Ü £ ¸$D0é´ê@».»©´ æ�¸�D0éaê@»bò�¸QÜ�¸�D¤é´ê@».»É-¤3å¸�S � ¸Qìs».» P °�¥Q8:,C(+3X£«�-� á"(+3'4�¥{8q,C('$ /:$êsëî¸Qº'ébï�»$�}£É(�¢7�(2�b�.-0¢n(27_4'/>¢8:¡��)8q�.-0=.-08:3m,b/�.9'70=�°2=.�'/}¡�/�(2�.9¨,./�÷�¢n(+3C*E/}/Á_�',./�b�./4�(+��÷"û w ã�ñ v÷�°2§s�'/>,./�ã�-0��=b�'/�M</*E/�.«q9'/¡�/�(2�.9¨,./m8:3"ìm° w ëHS}12¸{ìs»m¸�§s�¨-¤¢G��-0�]S}12¸{ìmé@ã)»b»s(23'4 v÷��<ã�$%5s�'/3�°¨÷ £ û w £ ñ v÷ £ °_§s�¨/,./ w £ û ¸�, £ µ®ÿw » n � ´ w-03\S}12¸{ìs»�(23'4 v÷�£"û ¸�,0£�µ ÿ v÷�» n � ´ v÷�¥{8q,�=b�'/C§}/n(+À���µ�=b8:�E8:708:«q£�8+¥_V�¸ ì}»m(2�T¢¹´ â̸�§s�¨/,./¬ÿw (23'4 ÿ v÷�(+,./j=.�'//Á_=./>3'�.-08:3¨��*_£�ºV82¥ w (23¨4 v÷�8:3�¶ ÛKP�»$M</>=]� g Q�(23¨4¤�ùëHS�?:¸b¸{º¨é.ï�»�U�ä1 � � ¸Qìs».»>$ÓB%Å_9)(+=.-08:3�¸�J'$�tq»�«q-¤�q/�

� HC �G¸{Ü £ » ð ¸$D0é´ê@»é@��¸�D0éaê@»�� �¹î�ï�ð @1� � � î�ï�ð @ Q2ê�ñ � HC � � ô�¸�æ £ ¸{è�éaê@»b».õ�Ü £ ¸{è�éaê@»�DõR�}¸{è�é´ê@»$Q_è�Q+êó � HC � � æ £ ¸{è�éaê@»�D�õR�}¸{è�éaê@»bò�¸QÜ £ ¸{è�é´ê@».»$Q_è�Q+êó � HC � � ø2¸Qè�é´ê@»$�}¸{è�é´ê@».ú4£)¸Qè�é´ê@» w £<¸{èE»$Q_è�Q+ê�ó � HC � � ø2¸{è�éaê@»��}¸{è�éaê@»bú0£<¸Qè�é´ê@» v÷�£<¸{èE»$Q_è�Q+êñ � HC � � Ü £ ¸Qè�é´ê@»$�}¸Qè�é´ê@».ö2¸Qè�é´ê@» w £ ¸{èE»$Q_è�Q2ê<ñ � HC � � Ü £ ¸{è�éaê@»��}¸{è�éaê@»bö2¸{è�éaê@» v÷ £ ¸Qè)»�QXè�Q+ê�ûÿºqI

¸�J'$0¹-J:»

¶a=}-¤�}Å_9'-0=./�/n(2�b£�=.8j�)(+�.�s=.8j70-0¡�-0=n°2(2�q¢<´ âÿ°_8q3�/n(2¢G��=./>,.¡ 82¥�¸�J¨$0¹�J:»�¸�9'�b-¤3¨«)°q¥{8q,��b8:¡�/s=b/,.¡��n°_=b�'/�4'8q¡�-03)(+=./4¢8q3X�q/,.«q/3'¢/j=.�¨/8:,b/¡�»}/Á_¢/>�'=s¥{8q,s=.�'/m7�(2�b=s=./,b¡­$%5s�'-0�s�',b82�:/��=.�)(+=s=.�'/V7k(+�.=}=./,b¡��)(2��(+70�.8�(j7¤-0¡�-¤=w§s�'-0¢@��§�/¢n(+7¤7.S�¸��%»$w5s�'/3�°¨8:3'/m�)(+�

� HC � Ü ð ¸�D0éaê@»>é*�}¸$D0é´ê@»$� �¹î ï�ð @ � � � î ï�ð @ Q+ê�ñ � HC � � ô�¸Þæ�¸{è�é´ê@».».õ�Ü�¸{è�é´ê@»�DnõR�}¸{è�é´ê@»$Q_è�Q+êó � HC � � æ}¸{è�éaê@»�D�õR��¸Qè�é´ê@».ò�¸{Ü�¸{è�é´ê@».»$Q_è�Q+êó � HC � � ø2¸Qè�é´ê@»$�}¸Qè�é´ê@».úE¸{è�é´ê@» w ¸Qè)»�Q_è�Q+ê<ó �

HC � � ø2¸{è�éaê@»��}¸{è�éaê@»bú)¸Qè�é´ê@» v÷%¸{èE»$Q_è�Q+ê

ñ � HC � � Ü�¸{è�éaê@»��}¸{è�éaê@»bö2¸{è�éaê@» w ¸Qè)»�QXè�Q+ê)ñ¯Ss¸���»}ûüº¸�J'$0¹�²:»

(23¨4

Ss¸��%»�û 70-0¡£�� T � HC � � Ü £ ¸Qè�é´ê@»$�}¸Qè�é´ê@».ö2¸Qè�é´ê@» v÷ £ ¸{èE»$Q_è�Q2ê�I_��b-03'«xM</¡�¡�(KJ¨$0¹�«q-¤�q/3�*)/>7¤82§ (+3'4�3'8:=b-03'«C=b�)(2= v÷��<ã�°'-0=s-0�s�)8q�.�.-0*'70/�=b8�¢G�'8_8:�./jÜ��.9'¢G�"=b�)(2=mÜ�ë¯S0T"¸b¸{º¨é.ï�»�US}12¸{ìmé v÷�»b»°EºKWåÜ�¸$D0éaê@»�Wä¹ v÷��.('$ /:$�¥Q8:,s('$ /:$�êsëî¸Qº'ébï�»�(+3'4Ss¸��%»�û � HC � � Ü�¸{è�éaê@»��}¸{è�éaê@»bö2¸{è�éaê@»�Q v÷�¸Qè)»�Q+ê�I

5s�'/3�°�¸�J¨$0¹n²:»w*E/¢8:¡�/�

á:®

Page 25: Modeling wells in porous media flows - CORE · Modeling wells in porous media ows Pierre Fabrie, Thierry Gallouet To cite this version: Pierre Fabrie, Thierry Gallouet. Modeling wells

� HC � Ü ð ¸$D0é´ê@»é@��¸�D0éaê@»�� �¹î�ï�ð @1� � � î�ï�ð @ Q2ê<ñ � HC � � ô�¸Þæ�¸{è�é´ê@».».õ�Ü�¸{è�é´ê@».DõR�}¸{è�é´ê@»$Q_è�Q+êó � HC � � æ�¸{è�é´ê@»�DõR��¸Qè�é´ê@».ò�¸{Ü�¸{è�é´ê@».»$Q_è�Q+êó � HC � � ø2¸{è�éaê@»��}¸{è�éaê@»bú)¸Qè�é´ê@» w ¸{èE»$Q_è�Q2êné>ó �

HC � � ø2¸{è�é´ê@»$�}¸{è�éaê@»búE¸{è�éaê@» v÷�¸{èE»$Q_è�Q2êné

ñ � HC � � Ü�¸Qè�é´ê@»$�}¸{è�é´ê@».ö2¸Qè�é´ê@» w ¸{èE»$Q_è�Q+ê)ñ �HC � � Ü�¸{è�é´ê@»$�}¸{è�éaê@»bö2¸{è�é´ê@»$Q v÷�¸{èE»$Q+ê�ûÿºqI

à -03'¢/��"-0��(+,.*'-0=b,@(2,b£C-03xS ? ¸.¸{º¨é.ï�»�U� 1 � � ¸{ìs»b»�¸Q(23'4x�m(+,.*'-0=.,G(2,b£C-03"¸�QEé@â ».»°+=.�'-0��£_-0/704'��¸Qá'$iá:ºq»$�}�8:=./�=.�)(+=n°q�.-03'¢/÷"û w ãmñ v÷�°_8q3'/x(+7¤�b8C�E(2�sÜ�ë\S0T�¸.¸Qº'ébï�»U@S}1n¸{ìméb÷�».»}(23¨4�ºZW Ü�¸$D0é´ê@»�Wä¹�÷��.('$ /:$·¥{8q,e(¨$ /q$<êsëO¸{º'ébï�»$�5s�'/,b/G¥{8q,./Ü��G(2=.-0�´Ä)/�s=.�¨/��./¢>8:3'4"�E(2,.=s8+¥e¸Qá'$0¹n²q»$�5s�'-0�s¢8:3'¢>7¤9¨4'/��=b�'/��',b8_82¥�8+¥�5s�'/>8:,./>¡ á'$0¹:$

á�¨�©«©«ª-þ�°�±³ø"¶-µ)ìèÕ@¶]¸(Àx´m¼Ø¶-ÀxÕ@´�ÇsÀfÅ(¶@ÅÚ¾m¶-µ.´�Â�¶ ÛZPuÐ�QRaü¹@Ôu¸�ÀqÅ�ï g ºqÒ¢ø"¶�µ�¸�÷�£)»)£�±�² ³ùÕ@¶�¸KÕ@´�ÇsÀfÅ(¶*ÅÚ¾¶@Ï-Ç�¶-Àfº@¶´$ÂÚÀf´(ÀfÀf¶1È�¸(µ�¿ÃÉ�¶�û0Àf¿Ãµ�¶�»u¶@¸�¾ÇsË$¶�¾u´(À"ì�¸�ÀqÅ�÷èÕ*¶R¸ÛÀf´�ÀfÀq¶kÈ(¸�µ�¿�É-¶�û0Àf¿�µ�¶R»u¶@¸�¾mÇsË�¶x´�À�ìÿ¾Ç�º@·«µ�·f¸(µ%÷�£Ñ´ ÷�Ê]¸�¾¢­´ âÌÊØÂ�´�ËKµ�·f¶R×0¶@¸-Ö�ê?µ«µ�´m¼Ø´�ß¹´@È�ýR´�ÂC¸�V�¸ vìs»b»1��Ò<ø"¶-µ�¸�Æ�£)»*£2±�² ³üÕ@¶u¸�¾¶@Ï�Çs¶-Àfº@¶R´$Â�ÂÇsÀfº-µ�¿�´(Às¾¢ÂË�´�» ìüíO¸Qº'ébï�»Kµ�´¶ Û;Çsº@·Hµ�·f¸�µÓ�ÒzÙ�·f¶-Ë$¶�¶*à(¿�¾µÃ¾<µ�×0´RÀf´�ÀqÀf¶1È�¸(µ�¿�É-¶�ÂÇsÀfº-µ�¿Ã´�Às¾ e Ê r ëHS0T"¸b¸{º'ébï�»U@V�¸ ì�»b»Y¾mÇsº@·Hµ�·f¸(µ e ¸$D0éaê@»�W�Æ £ ¸$D0éaê@»�W r ¸$D0é´ê@»÷ £ ê$¸(Ò�¶-Ò�Ê�´(Ë<¸(Ò�¶-Ò�ê}ëî¸Qº'ébï�»K¸(ÀfÅ]´(ËK¸�ßÃß�¢"ë�¶ }ÚÊ

Ñ�Ò � HC � � ��¸Qè�é´ê@»�Æ�£<¸Qè�é´ê@»$Qq÷�£<¸{èE»$Q+ê<·f¸�¾K¸Rß¹¿Ã»u¿�µ�Ð$¿ÃÀC¶ ÛYÔ�Ê¢¸�¾�¢<´ âùÊf´(ËK¸�ßÃß��OëHV�¸ vì|í­¬iº¨é.ï}¯{»ÒÙ�·f¶-ÀsÊ¢µ�·f¶-Ë$¶Z¶*à(¿�¾µ�¾�ÆjëHS0T�¸.¸{º¨é.ï�»�U*Se1n¸{ìméb÷�».»l¾mÇsº@·Hµ�·f¸(µ

� HC � � ��¸Qè�é´ê@»�Æ £ ¸Qè�é´ê@»$Qq÷ £ ¸{èE»$Q+êx´ � HC � � ��¸Qè�é´ê@»�ƨ¸{è�éaê@»�Qq÷�¸{èE»$Q+êné�o��OëHV�¸ vì|í­¬iº¨é.ï}¯{»IR"Ç�Ë�µ¹·q¶�˻ڴ�Ë�¶Ê0´(Àf¶<·q¸�¾ e ¸$D0é´ê@»0W�Æ'¸�D0éaê@»�W r ¸$D0éaê@»�÷�ê$¸�Ò�¶-Ò�Êf´(Ë<¸(Ò�¶-Ò�ê}ë�¸{º¨é.ï�»�Ò��¬�ç�ç2y]ç�y�á�¨s©«©«ª¹þE°�±M</>=���û ì í�¸{º¨é.ï�»>$�&¨8:,©¢�ë�¶ }�°�70/=u£�£­*E/�=b�'/�¡�/�(2�.9¨,./�8:3�� 4'/GÄE3'/4�*_£«Qs£�£)¸Qè�é´ê@»mû�Qq÷�£<¸{èE»$Q+êC(23¨4O7¤/>=£O*E/�=b�'/�¡�/n(2�b9',./�8q3�� 4'/GÄ)3¨/4 *X£«Qs£�¸Qè�é´ê@»mû�Qq÷�¸Qè)»�Q2ên$�5s�¨/��./>ÅX9¨/3'¢/�¸�£�£'»*£2±�² ³ä-0�C(�*)8q9'3'4'/4��./Å_9'/>3'¢/82¥}3'8:3'3¨/«X(+=.-0�:/j¡�/n(+�.9',b/��8q3���(+3'4«£�£¹´ £�°�(+�©¢A´ âü°¨¥{8q,x=b�'/�§}/n(2À(��µ�=.8q�)8q708:«:£�8+¥É¸�V�¸ v�m».»=�a$ &¨,.8q¡�=.�'/Ä),b�.=m-0=./¡ 82¥�=b�'/C�_£_�E8:=.�¨/�./�V8:3 ¸ZÆ�£'»*£2±�² ³�°<8q3'/C4¨/4'9'¢/>��=.�)(+=©Æ�£­ë¯S 1 ¸���é*£�£X»°�¥{8:,m(2707�¢OëO¶ }�°�(+3'4"=.�E(2=m=.�'/�./>ÅX9¨/3'¢/�¸ZÆ £ £ £ » £�±�² ³ -0�s(j*)8q9'3'4'/>4��b/Å_9'/3'¢>/C8+¥½¡�/n(2�b9',./>�s8:3 ��$_��b-03'«"=.8�=b�'/��b/¢8:3¨4O-¤=b/¡ª82¥s=.�'/��_£_�E8:=.�¨/�./��8q3 ¸�Æ £ » £2±�² ³ °�=.�'/��b/Å_9'/3'¢>/î¸�Æ £ £ £ » £�±�² ³ ¢8:3_�q/,.«q/�C=b82§e(+,.4'�j(¡�/�(2�.9¨,./ v£�8:3 v� ¸k¥{8:,s=b�'/m§�/n(+À(�Zµ�=b8:�E8:708:«q£C8+¥�V�¸ v�m».»>°'=.�E(2=s-0�x¸Þ§s-0=.�H^Cû ¸{è�éaê@»b»

�!��¸�^:»�Æ £ ¸�^:»$Qs£ £ ¸�^q»�´ �

"!��¸�^:»�Q v£�¸�^q»é7op�OëHV�¸ v�x»�I

5s�'/j/Á_-0�.=./>3'¢/�82¥qÆ�ë¯S 1 ¸���é@£)»}�b9'¢G�O=.�)(+=<Q v£<¸�^:»�û ƨ¸�^:»�Qs£�¸�^:»�¥{8q707¤82§s�s8+¥�=.�¨/ (+*'�.8q7¤9¨=./j¢8:3_=.-03_9'-0=a£�8+¥ v£�§s-¤=b�,./>�.�E/¢=m=.8R£"¸Þ=.�)(+=s-¤� v£��T��£�°q§s�'/,./Ú£�-0�s¢8:3¨�.-04'/,b/4"(2��(V¡�/n(+�.9',b/�8:3 v� §s-¤=b�L£�¸ v�Aé���»%ûÿºq»}=.�)(+=�§}/m�',.82�q/3'82§�$¶a3"8q,.4'/,�=.8��',.82�q/�=.�)(+= v£¬���³£�°'70/=sý *)/�(j*E8:,./>7¤-�(+3��b/=m82¥ v�Ú�b9'¢G�"=.�)(+=<£�¸�ým»�û º¨$ r 3'/V�)(2��=.8��¨,.82�:/V=.�)(+=v£<¸Þýx»sû º¨$]M</>=$# �üý *E/ (j¢8q¡��)(+¢=��b/=n$�M</>=�Ö g º'$�&',b8:¡ =.�'/j,./«q9'7�(2,b-¤=a£�82¥�=b�'/j¡�/n(+�.9',b/u£�°¨5s�'/,b/C-0�x(+38:�E/3��./=&% 82¥ v� ¸�=b�)(2=}-0�'%äû v�A�)(�§s�'/,b/&(�-0�s(23�8q�)/3��b/=�8+¥�¶ Û P »w�:/,b-ã¥{£_-¤3¨«xý �*%Ú(23¨4L£�¸�%j»0WmÖ_$%5s�'/,.//Á_-0�.=b�x(V¢8:3_=.-03_9'8q9'�}¥{9'3'¢>=.-08:3H��¥Q,.8:¡ v�ä=.8�¶ Û O �b9'¢G�"=.�)(+=

�}¸�^:»}û ¹:é(op^jë+#"é��¸�^:»}ûÿº¨é(op^�ë,% � é�}¸�^q»}ë�¬iº'é�¹G¯.é-op^�ë v�`Iá2±

Page 26: Modeling wells in porous media flows - CORE · Modeling wells in porous media ows Pierre Fabrie, Thierry Gallouet To cite this version: Pierre Fabrie, Thierry Gallouet. Modeling wells

5s�'/3�°'8:3¨/x«q/=.�

v£<¸�#"»_W �"!�}¸�^:»$Q v£�¸�^q» W 70-0¡£-� T � ! ��¸�^:»�Æ £ ¸�^:»$Qs£ £ ¸�^:»W 70-0¡£-� T � ! ��¸�^:» r ¸�^:»�Qs£ £ ¸�^:»W �

!�}¸�^q» r ¸�^q»$Qs£�¸�^:»YWùcè£�¸�%j»0WùcEÖ_é

§s�'/,b/xc�û � � � ��� "! � � r � � ÷ ��� C � H ��ø �7� �����G$m5s�'-0�m«:-0�:/��°<�.-03'¢>/®Ö g º�-0�m(2,.*¨-¤=b,@(+,.£_° v£<¸�#�»�ûÚº'$m5s�'/>3�° v£·¸�ýx»�ûÚº¥{8q7¤7082§s��¥{,.8q¡ =b�'/�,./>«:9'7�(2,b-0=a£­8+¥e=b�'/�¡�/n(2�b9',./ v£ ¸Q(23'4O=b�'/j¥ (2¢>=�=.�)(+=.#L-0�C(+3_£"¢8:¡��)(2¢>=x�b9'*'�b/=�82¥sýx»>$�5s�'-0�¢8q3'¢709'4'/��=.�'/V�',.8_82¥%82¥ v£����ù£�(23'4�«q-¤�q/�}=.�'/�/>ÁX-0�b=./3'¢>/C8+¥xÆjëHS}12¸���é@£)»w�.9¨¢@��=b�)(2= v£�ûEÆ�£�°_=.�)(+=s-0�

�"!�}¸�^q»$Q v£�¸�^:»wû � HC � � �}¸{è�é´ê@»�Æ'¸Qè�é´ê@»$Qq÷�¸Qè)»�Q2ênéEop�ùë¤V�¸ v��»�I

¶a=s,./>¡�(2-03'�}=.8��¨,.82�:/m=b�)(2= e ¸$D0éaê@»�W�Æ'¸�D¤é´ê@»�W r ¸�D0éaê@»}÷��.('$ /:$0°+¥{8:,�(¨$ /q$<êsëO¸{º'ébï�»$à -03'¢/ e ¸$D0é´ê@»�W�Æ £ ¸�D0éaê@»7W r ¸�D¤é´ê@»}÷ £ �.('$ /:$0°+¥{8:,�(¨$ /q$�ê�ë�¸{º¨é.ï�»}(23¨4�¥Q8:,s(2707�¢�ë�¶ }�°_8q3'/��)(+�n°q¥{8q,�(2707p�ùëHV�¸ v��»>°��3'8q3'3'/«_(2=.-0�q/:°

� HC � � e ¸Qè�é´ê@»$�}¸Qè�é´ê@»$Qq÷ £ ¸Qè)»�Q+ê¢W � HC � � Æ £ ¸{è�éaê@»��}¸{è�éaê@»�Q:÷ £ ¸{èE»$Q+ê�W � HC � � r ¸Qè�é´ê@»$�}¸Qè�é´ê@»$Qq÷ £ ¸{èE»$Q+êné§s�'-0¢G��£X-0/704'��°_9'�E8:3��E(2�.�b-03'«�=.8j=.�'/V7¤-0¡�-¤=�(+�â­´ â�(+3'4�9'�b-03'«�=.�'/m4'8q¡�-03)(+=./4�¢8q3_�:/,b«:/3'¢>/�=.�'/>8:,./>¡ ¥{8:,}=.�'/-03_=./«q,@(2=b-08:3�-03�=.-0¡�/s-¤3�=b�'/�Ä),.�b=x(23¨4�=.�¨-¤,b4�-¤3_=b/«:,G(270�n°

� HC � � e ¸Qè�é´ê@»$�}¸Qè�é´ê@»$Qq÷�¸{èE»$Q+ê�W �HC � � ƨ¸{è�éaê@»��}¸{è�éaê@»�Q:÷�¸Qè)»�Q+ê¢W � HC � � r ¸{è�éaê@»���¸Qè�é´ê@»$Qq÷�¸Qè)»�Q2êné

¶a=}¥{8q7¤7082§s�}=.�)(+= e W¸Æ`W r £s�.('$ /:$0°q§s�'-0¢@��-0��/>ÅX9¨-¤�:(+70/3_=�=b8 e ¸�D0éaê@»�W�Æ'¸�D0éaê@»�W r ¸�D0éaê@»}÷��.('$ /:$½¥{8:,�(¨$ /q$%êsë�¸{º¨é.ï�»>$5s�'-0�s¢8q3'¢709'4'/��=.�'/m�',b8_82¥�82¥0M</¡�¡�(�J'$0¹:$M</>=s9'��«q-¤�q/�}3'82§ÿ=b�'/m�.Àq/=.¢G�"8+¥½=b�'/m�',.8_8+¥½8+¥�5s�'/>8:,./>¡ á¨$ áj(23¨4�5s�¨/8:,b/¡ á¨$i®'$��­�¨sæ��-ö¯ç2y<æ-ö�¨<��¬�ç�ç�y]ç2ylõlö|¨sç�¬�¨s© ®E°¹®Â��É¥{8:,�=.�'/��',b8_82¥s82¥�=b�'/�#%,.8q�)8q�.-0=.-08q3îá¨$¤¹q°E§�/�*'9¨-¤704�°¨¥{8q,m÷�«:-0�q/3­-03\Sw?:¸Qìs»°%(�,./>«:9'7�(2,V�.8:709'=b-08:3"=b�',.8q9'«:��=.�'/à ¢G�)(+9'4'/,�Ä)Á_/4��)8q-¤3_=s5s�'/>8:,./>¡ §s-0=.��=.�'/j(2�'�'70-0¢n(+=.-08:3x¦}¸{ø>»�ûüø N 4'/ Ä)3'/4­(+�}¥{8:707082§äç&¨8:,s(23_£�«q-¤�q/3�ø+°'=.�¨/,./�/>ÁX-0�b=.��(j9'3'-0Å_9'/m�.8q709'=.-08:3�¸Tß�é.æ%»�8+¥

æ�¸{è�é´ê@»)ñ + ¸Qè)»£�¸{ø2¸Qè�é´ê@».» ¸{õmß�¸{è�éaê@»·ó¹,E¸{ø2¸Qè�é´ê@».»´þ�»�ûüº4'-0�Xæ}¸{è�éaê@»·ó�úE¸{è�éaê@»´÷�¸{èE»<ñ�ö2¸Qè�é´ê@»a÷�¸Qè)»%ûüºæ�¸{è�é´ê@».D�F­ûÿº¨é � � ß�¸Qè�é´ê@»$Q_è�ûåºV¥{8q,�('$ /:$·êsëî¸Qº'ébï�»I

5s�'/3�°'=.�¨/,./�/>ÁX-0�b=.�x(V9'3'-0Å_9'/�ø N �b8:709'=b-¤8q3�8+¥ø N ð ¸{è�éaê@»<ñ�4'-0�<¸�æCø N »>¸{è�éaê@»·óO4'-0� ' ¸�ã�¸{ø>»�ñ�ô�¸Þæ�».»bõ�ø N ( ¸Qè�é´ê@»ñmø N ¸{è�éaê@»búE¸{è�éaê@»´÷�¸{èE»<ó vø2¸{è�éaê@»bö2¸{è�é´ê@»a÷�¸{èE»%ûåº'é¸�ã�¸{ø2¸Qè�é´ê@».»�ñ�ô�¸�æ}¸{è�éaê@»b».»_õ�ø N ¸{è�éaê@»�D@FOûåºø2¸{è�éGº:»%ûÿø C I

5s�'/3�°'=.�¨/x«q/3'/,G(27<¢n(+�./�¢>8:3'�b-¤4¨/,./4�-¤3�5s�'/8q,./¡�á'$iám-0�s8:*¨=@(2-03'/>4�*_£��)(2�b�.-03'«�=b8C=b�'/m7¤-0¡�-¤=%8q3��.¡�8_8:=.��(+�'�',b82Á��-0¡�(+=.-08:3'�w82¥�=b�'/m¡�/n(+�.9',b/m÷�$%5s�'/,./j(2,./m3'8�(+4'4'-0=.-08q3)(27E4'-ãâ�¢9'70=.-0/�s=b8C=b�'/��',b8_82¥�8+¥�5s�'/8:,b/¡�á'$0¹:$

��­�¨sæ��-ö¯ç2y���¬�ç�ç2y½õ�ö�¨sç�¬�¨�©÷®E°ML/ #%(2,G(2*E8:70-0¢�,./>«:9'7�(2,b-0�@(2=b-08:3

á�J

Page 27: Modeling wells in porous media flows - CORE · Modeling wells in porous media ows Pierre Fabrie, Thierry Gallouet To cite this version: Pierre Fabrie, Thierry Gallouet. Modeling wells

¶a3�8:,b4'/,�8+¥�(��:8:-04�=.�'/�4¨/«:/3¨/,@(+¢£"82¥%=.�'/É¥{9'3'¢=b-08:3H¡"§}/m-03X=b,.8_4'9'¢>/C(j¡�8_4'-ãÄ)/4��',.8q*'70/¡ §s�'/,./Ú¡"-0�s,./>�'7�(2¢/4-03"¸{á¨$ ®qµ:»%*_£R¡ 0�ûÁ¡�ñ¬Öäé�Ö g º'$�5s�'-0�}ÀX-03'4�8+¥�,./«q9'7�(2,b-¤¼�(2=.-08q3�,./4'9¨¢/��=b�'/��¨,.8:*¨7¤/>¡�82¥�-0¡�¡�-T�b¢-0*'70/%¦)82§å=b8�=.�'/¡�8_4'/>7)�',b8:*'70/¡ª¸{á¨$0¹n»��¸{á¨$iá:»°+¥{8q,�§s�'-0¢G��=.�'/>,./�-0��(2=s70/n(+�.=s8q3'/�¸�,./>«:9'7�(2,G»}�.8q7¤9¨=.-08:3�¥{8:,s÷�«:-0�:/>3�-03LS ? ¸{ìs»>$/ _�3¨-T¥Q8:,.¡Ú/�b=.-0¡�(+=./�s-03�=.�¨/x,b/«:9¨7k(+,s¢n(2�b/�}£"=@(+ÀX-03'«zìO¸�i�»mû � íC ¡�¸��<»�Qs�O§}/C�',b82�:/j§s-0=.�'8q9'=�(23_£�4'-ãâ�¢9'70=a£�=.�)(+=x=b�'/,b/�/Á_-¤�b=m=a§�8�¢8q3X=b-03X9¨8:9'��¥{9'3¨¢=.-08:3¨�e 1 ¸0ê@»s(+3'4 e ? ¸0ê@»>°'-03'4'/�E/3'4¨/3_=�82¥GÖC�b9'¢G��=.�)(+=s=.�'/��b8:709'=b-¤8q3Li�0m82¥�=b�'/m,./«q9'7�(2,b-¤¼>/4"�',b8:*'70/¡��q/,.-ãÄ)/>�

� ðC � õRìO¸�i�0:¸�D¤é Y'»b» � ?� � �B�p� Q[Y`W e 1 ¸kê@»é ¸�J'$0¹-[:»� ðC � Gqð1i 0 ¸ZY'» � ? � ï � �p��� � Q[YRW e ? ¸0ê@»�I

5s�'/3�(2�b�.9'¡�-¤3¨«x=b�)(2=�ì-��1�-0�e(+3J�C;8q7¤4¨/,�¢>8:3_=.-03_9'8:9¨�%¥{9'3'¢>=.-08:3�§s-¤=b��¡�8_4'9'709'�G,E°_§�/�4'/4'9'¢>/�¥Q,.8:¡Ú=b�'/��',b/�_-08:9'�/�b=.-0¡�(+=./C¸Þ�.//j/:$ «)$}¬0¹�[>¯{»}=.�E(2=]i 0 -0�s*E8:9'3¨4'/4�-03LS�?�1�2 ' ¸Qº'ébï�»U��4352 � ?�1�22¸{ìs» ( ¥Q8:,s(23_£ 6}éGºA�76«� ¹q$5s�'/�b/�/�.=b-¤¡�(2=b/� (+,./��.9_â�¢-0/3_=j=.8"«q/=�(��b8:709'=.-08q3¯i�8+¥C¸{á¨$ ®qµ:»k� ¸Qá'$ ±Xáq»��¨,.82�_-04'/4�=b�)(2=V=.�'/�/Å_9)(+=.-08:3å¸Qá'$i®:µq»m-0�§s,.-0=b=./3�8:3�=b�'/m¥{8q7¤7082§s-03'«�¥{8q,.¡

i ð ¸{è�éaê@»Eñ�4'-0�x¸Þæ H ò�¸�i�».»>¸{è�éaê@»wó­4¨-¤�<¸QõRìO¸�i�».»>¸{è�éaê@»b»%ó­4¨-¤�<¸Ã­�¸�i�».»¸Qè�é´ê@»ñ�iq¸{è�éaê@»bú)¸Qè�é´ê@»a÷�¸Qè)»�óOö2¸{è�é´ê@»a÷�¸{èE»%ûÿºqI

à�8q,./82�q/,�(2�Ri�-0�C9¨3'-ã¥{8:,b¡�70£"*E8:9'3'4¨/4 -¤3~S T ¸b¸{º¨é.ï�»�í�ìs»°w=.�'/�/�b=.-0¡�(+=./�¸�J'$0¹-[:»m,b/¡�(+-¤3¨�C�:(+7¤-04�$�5s�'/�8q3'70£4'- A</,./3¨¢/C-0�s=b�'/m/�.=b-¤¡�(2=b/�8:3�=b�'/m=.-0¡�/�4'/,.-0�:(+=.-0�:/�§s�'-0¢@��*E/¢8q¡�/�

� ðC � G ð i80:¸�D0éVY'» � ? �¹î ï�ð @ �B����� � Q[Y`W e � ¸kê@»éÛop� g Q�I5s�'-0�s/3'4¨��=.�¨/x�¨,.8_82¥�82¥%5s�'/8q,./¡�á'$i®'$

9���dq�s���m���s���¬0¹ ¯$:);8<>=�?�@�AB@0°�CED�FHGJILKNMEO�=QPî(+3'4SR)D�O8DUTL<LDVT>=�:�@0°:^�À«Àf´�Às¾µ�¸�µ�¿�´�Àq¸�Ëý�Æ�´�×èµ�·fË$´(Ç�È�·`¼f´�Ë$´(Ç(¾X»Ú¶@Å�¿�¸q$)Â�3'3�$à"(+=n$)#½9',@(jÂ��'�'7a$<¸�±_»}�:8:7�¹�®:²¨°)¹-t:µ2±E°_�'��$'®qº:®-�.®'¹�²'$

¬iᳯ$:);8<>=J?�@NAB@_(23'4XWEDZY'[\TL[ WE[ <L<E]�=J^$@k°�²Z´�Às¾µ�¶*¸(Å�ý�Æ¢´(×~´$Â]×�¸�µ�¶�ËY¸�ÀfÅR´(¿�ßpµ�·fË�´�Ç�È-·z¿�À�·f´(»<È(¶-Àf¶@´�Ç(¾�¼Ø´(Ë$´�Ç(¾»Ú¶*Å(¿�¸:$¨Â�3'3�$ à ¢9'8q7k(Ú}�8:,b¡­$ à 9'��$E#%-0�@('°<¸Þ±X»>°_�:8:7�¹�á'°_3'8�®¨°<¹�tqµ�J'°q�'��$E®:®(J-�.®�t:á¨$¬i®³¯�Y_KL`L[ TLaH;bK�<L<dc @Hef@V=g^>TL<E]Hh�R�@Hi*@_(+3'4XjEkmlnILDVG�R @oi*@0°[R�ß¹´(׫´$Â.Æ�Çs¿�Å�¾]µ�·fË�´�Ç�È-·xÀf¸(µ�ÇsË�¸�ß�Ë�´�º@Ö�¾$qpm709_§�/>,Â�¢�(24'/¡�-0¢É#½9'*'7a$0°)¹-t�t:º¨$

¬ ±+¯$r�@)Y'[\KL`7rm@�(23'44R)[ `L`HMLDVs8<t: @0°vu\´-Å�¶�ß¹¿ÃÀ�È«È�Ë�´�ÇsÀfÅ�ê$×0¸�µ�¶�ËZÆ¢´(×�¸(ÀfÅh¼f´�ß�ß�Ç�µ�¿�´(À'$Ó%j$�Û�/-04'/7�#%9'*�$_b}8E$0°%�8:,b4',./¢G�_=n°%¹-t:µ�[¨$¬�J³¯�Y&]EFLFHKL`LW�]BC'@s(+3'4cwKL;b;-]gM_x[ <zy&I>@k°]²K´(Àsß¹¿�Àq¶*¸(ËH¶�ß�ß¹¿ ¼�µ�¿ÃºL¸(ÀfÅۼظ(Ë$¸(Õ*´�ß¹¿�ºÛ¶*Ï-Çs¸�µ�¿�´�Àؾ\¿�ÀfÉ-´�ß�É�¿�À(ȯ»Ú¶¾ÇsË$¶Å�¸(µ�¸q$ {E$)&¨9'3'¢=�$'Â�3E(27a$0°:�q8:7a$Xµ(['°<¹-t:µ�t¨°_�'��$<¹>±�t-�¹n²(t'$¬i²³¯�|&ILK}hL[ TL<Bc @�(+3'4~r KL����`L[�rm@0°�uh¸�µ�·f¶-»u¸(µ�¿�º@¸�ß]»u´-Å�¶�ßB¾z¸�ÀfÅ�û0Àf¿�µ�¶H¶�ß¹¶-»u¶-ÀfµÃ¾<´(ËLË�¶¾¶-ËÉ-´�¿�ËÚ¾¿�»ÚÇ(ß¹¸�µ�¿�´(ÀfÒþ�¿�À(È�ß�¶Y¼Ø·f¸�¾¶Ê¢»ÚÇ(ß¹µ�¿ ¼Ø·q¸�¾¶Ú¸�ÀfÅz»ÚÇ�ß�µ�¿Ãº*´(»�¼f´�Àq¶�Àfµ�Æ�´�×�¾Kµ�·fË$´(Ç�È�·`¼f´�ÇsË$´(Ç�¾K»Ú¶@Å�¿�¸q$ à =b9'4'-0/�s-03�¡�(2=b�'/¡�(2=.-0¢�(23¨4�-0=b��(2�'�'70-0¢n(+=.-08:3'��°:�q8:7a$<¹�[¨°s}�8q,.=.�Ø�Q��8:707�(23'4�#%9¨*'70-¤�b�'-03'«xb}8:¡���$0°'¹�tqµ:²¨$

¬�[³¯��&K}��Of]gT�|.@\��@q(23'4�A�IL[ [ ;}[ `�i*@o��@0°.Ù�¿�»Ú¶�¾�¼�ß�¿�µ�µ�¿�À�È�»Ú¶-µ¹·q´�Å0Â�´�Ë�¸�Å�É-¶@º�µ�¿�´(Àsê$Å�¿ ��Ç(¾¿�´(Àsê$¶@Ï�Çs¸�µ�¿�´(ÀL¸(Ë¿�¾¿�À�È¿�ÀRº@´�Àqµ�¸(»u¿�Àq¸�Àfµ�µ�Ë$¸�Àؾ�¼f´�Ë�µ{$>¶$b}¶aÂxàÍt'¹:°2#%,b8_¢//4'-03'«V82¥)=b�'/ à /¢8:3¨4xb}8:3_¥{/>,./3'¢>/s8:3�¶a3'4¨9'�.=b,.-�(27'(23¨4CÂ��'�'70-0/4à"(+=.�'/¡�(2=b-¤¢>�n°_�'��$f[¨¹@�.µ:ᨰ'Ûs8:*E/,b=�Bs$ r ©ià"(27070/£_°b{q,n$_/4¨-¤=b8:,�° à ¶aÂmà #½�'-07k(+4'/70�'�'-�(¨°)¹-t�t'¹q$

¬iµ³¯�cwKL�ETL[\ML��c =�(23'4�iSKLWLKHMLTL[ �8y�]L`�<�i*@0°-D�Àf¸�ß�ý-¾¶è»u¸(µ¹·��¶�»Ú¸�µ�¿�Ï-Ç�¶èÅ(¶è»u´-Å��¶�ß¹¶�¾³Àf´(ÀÍß¹¿�À��¶�¸(¿�Ë$¶�¾ùÅ�¶ßV» ¿�À�È_�¶�Àf¿Ã¶�˿ö�¼��¶-µ�Ë�´�ß�¿��¶�Ë�¶n$_à"(+=.�_�/¡�(2=b-¤Å_9'9¨/��/>=�Â��¨�'70-¤¢�(2=.-08q3'�eáqá'° à �¨,.-03'«:/>,1����/,.7�(+«)°<¹�t(t:²¨$á:²

Page 28: Modeling wells in porous media flows - CORE · Modeling wells in porous media ows Pierre Fabrie, Thierry Gallouet To cite this version: Pierre Fabrie, Thierry Gallouet. Modeling wells

¬�t³¯�cw` x]E�J[ `�� @0°�DÁ�ä1 � ásê$¶¾µ�¿�»Ú¸�µ�¶�´�Ë�¾´�ß�Ç�µ�¿�´(Às¾¢µ�´K»u¿ à�¶@Å�Õ@´�ÇsÀfÅ�¸(ËýKÉ-¸�ß�Ç�¶7¼pË�´�Õ�ß¹¶-»K¾�Â�´�Ë�¾m¶@º@´�ÀqÅK´�Ë�Å�¶-Ë¢¶�ß�ß¹¿ ¼pµ�¿�ºÅ�¿ ��¶�Ë�¶�Àfµ�¿�¸�ß|¶@Ï�Çs¸(µ�¿�´(Às¾° à �¨,.-03'«:/>,1����/,.7�(+«)°'à"(+=.��$_Â�3'3�$)áqµ:®¨°<¹�t:µ(tm�'�"²�[(t-�.²:µ(['$¬0¹nº³¯�CEKLWHkmln[ TEO8GJKLs8K�� @�: @V=ZPN]g;-]gTLTLDVGE]Hh�R�@�:�@)(+3'4���`LKL;���FH[ hbK���@���@0°�ø�¿�Àf¶@¸�ËZ¸�ÀfÅzÏ-Ç�¸�¾¿�ê1ß¹¿�Àq¶*¸(ËK¶@Ï�Çs¸�êµ�¿�´�Às¾K´�Â�¼Ø¸(Ë$¸(Õ*´�ß¹¿�ºZµ�ý�¼f¶n$¨Â�¡�/,.-0¢n(+3�à"(+=.�'/¡�(2=b-¤¢�(27 à 8_¢-0/=a£X°<¹-t:²qµ'$¬0¹:¹ ¯�CED�]gTEO�r�@JC'@w°���Ç�¶�ß¹Ï-Çs¶¾K»~�¶�µ�´-Å�¶�¾lÅ(¶uË��¶¾´�ß�Çsµ�¿�´(ÀHÅ�¶½¼pË�´�Õ�ßf�¶-»u¶�¾<¸(Ç�à�ß¹¿�»Ú¿�µ�¶¾]Àf´(ÀHß�¿�À��¶-¸�¿�Ë�¶¾$[%�9'3'8_4�°<¹�tq²�t¨$¬0¹nᳯ�iS[�kE[ `EO���@�c @0°�D�ÀHSE᯶�¾mµ�¿�»Ú¸�µ�¶]´(Ëuµ�·f¶ÚÈ�Ë�¸�Å(¿�¶�Àqµ�´$ÂÚ¾´�ß�Ç�µ�¿�´(À«´$ÂZ¾m¶@º@´�ÀqÅ\´(Ë$Å�¶-ËÚÅ�¿�É-¶�Ë�È�¶-Àfº@¶x¶@Ï-Çs¸�µ�¿�´�Às¾°

Â�3¨3�$ à bm$s}�8:,b¡­$ à 9'��$E#%-0�@('°<¹-['°0¹�tq²:®¨°:�'�O¹�µ�t-�.á:ºq²'$¬0¹n®³¯�i�]gTLDV[ `~:�@0°�´�ÀHµ�·f¶ÚË$¶1È(Ç(ß¹¸�Ë�¿�µ�ýÚ´$ÂZµ¹·q¶K¾´�ß�Çsµ�¿�´(ÀH´$Âl¾m´(»u¶Ú¶�ß�ß¹¿ ¼�µ�¿Ãº<¶@Ï-Çs¸�µ�¿�´�Às¾$ à 9'*'¡�-¤=b=./4��)(2�E/,�$¬0¹±+¯��>`LD �ETL[\<�: @0° ¡½¶�»Ú¸�Ë$Ö�¾<´(ÀH¶@à�¿�¾µ�¶�Àqº*¶Ú¸�ÀqÅxÇsÀf¿�Ï-Çs¶�Àf¶�¾@¾u´$ÂY¾´�ß�Çsµ�¿�´(Às¾K´$Â�¶�ß�ß¹¿ ¼�µ�¿Ãº�¼pË�´�Õ�ß¹¶-»K¾�×0¿�µ�·zË¿BÈ-·fµÃê�·f¸�Àqž¿�Å�¶Ú»Ú¶*¸�¾Ç�Ë�¶¾$'Û�/3'4�$Eà"(2=�$_Â��'�'7a$<¸�[:»m¹�J¨°_3'8)$'®¨°0¹�t�t(J'°q�'��$'®qá'¹@�.®:®(['$¬0¹�J³¯��>`LD �ETL[\<¢:�@0°RÜ0´(ÀfÅ�¿Ãµ�¿�´(Às¾\¸(Ç>à ß�¿�»Ú¿�µ�¶�¾HÀf´�À~·f´(»u´@È��¶�Àf¶�¾�¼Ø´(Ç�ËhÅ�¶�¾R¼�Ë$´�Õ�ßf�¶-»Ú¶¾H¶�ß�ß¹¿ ¼�µ�¿ÃÏ�Çs¶�¾¤¸(É�¶@ºh¾¶@º@´�ÀfŻڶ�»ÚÕ�Ë�¶u»Ú¶�¾mÇsË�¶n$_Â�3'3�$)&E(2¢q$ à ¢-a$_5%8:9'708:9¨�./x²¨°'3'8E$'á'°0¹�t(t�[¨°2�'��$Eá�t�[>�´®¨¹nµ¨$¬0¹n²³¯�P [ `L`LDVT£r�@0°G·�¸�µ�·f´�ß¹´@È�¿Ãº*¸�ߢ¾´�ß¹Çsµ�¿Ã´�Às¾R´$Â�¶ß�ß�¿ ¼pµ�¿�º�Å�¿ ��¶-Ë$¶-Àfµ�¿�¸�ߢ¶@Ï-Çs¸�µ�¿�´�Às¾$�Â�3'3�$ à ¢>9'8:7�(z}�8:,.¡�$<#%-0�@('°}¹�tq²2±E°

�'��$E®:µ(J-�.®:µ�[¨$¬0¹�[³¯�P DV¤�]LT�r�@0°-¡½¶kÈ(Ç(ß¹¸(Ë¿�µ �¶<Å(¶�ß¹¸Rº@´(»�¼f´�¾��¶-¶KÅ�¶KÅ(¶�Ç�à�Â�´�Àfº-µ�¿Ã´�Às¾�¶�µ0¸m¼�¼�ß¹¿�º@¸�µ�¿�´(Às¾$��}8:707a$�_�3�$Xà"(+=n$q¶a=@(+7 $�� ¸�J:»�¹n²¨°

3'8E$)á'°<¹-t�[(t'°q�'��$�Jqº'¹m�=Jqá:á'$¬0¹nµ³¯�P DV¤�]LT~rm@k°�Ü0´�»½¼Ø¸(º�µ�¾¶�µ�¿�ÀRS�á<¸{º¨é.ïKU ��»>$_Â�3'3�$'à�(2=n$E#%9',G('$_Â��¨�'7a$¤°_¶��j°�¹>±X²'°E¹�tqµ�[¨°_�'��$'²�J>�=tq²'$¬0¹�t³¯�P <�KL¤&¥qKLFLFHILDVK�c @0°�ø.¶l¼pË$´(Õßf�¶�»Ú¶ÚÅ�¶uúZ¿�Ë¿�º@·sß�¶�µE¼f´�ÇsËÚß¹¶�¾¦�¶*Ï-Çs¸�µ�¿�´�ÀؾX¶�ß�ß¹¿ ¼pµ�¿�Ï�Çs¶�¾uÅ�Ǥ¾m¶@º@´�ÀqÅH´�Ë�Å�Ë$¶~�¸zº@´-¶ZN êº-¿�¶�Àqµ�¾KÅ�¿�¾mº@´(Àfµ�¿ÃÀfÇ(¾ $¨Â�3'3�$'¶a3'�b=n$)&¨8:9¨,.-0/,n°'6É,./3'8q*'70/:°'�q8:7a$E¹�J'°<¹-t:²(J'°q�'��$�¹�µ�t>�´á(J:µ'$¬iá:º³¯�y&`L]LDVKLTLDU[ ;b;�]~c @gi~@0°fÎ�ß�ß�¿ ¼pµ�¿�º¢Î½Ï�Çs¸�µ�¿�´(Às¾<¸(ÀfÅC^�Õ�¾mµ�¸�º�ß¹¶q·�Ë�´�Õ�ß¹¶-»K¾$25s�¨/�9'3'-0�q/,.�b-¤=a£j�./>,.-0/�s-03C¡�(2=b�'/¡�(+=1�

-0¢�n°E#%70/3_9'¡�#%,./�b�n°�}�/>§~§�8q,.À<°<¹�tqµ�[¨$

á�[