Assignment 7
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Assignment 7 – Mark Thomas CID: 00645751 Isolated inverted 5-spot pattern Isolated →q 1 =q 2 +q 3 +q 4 + q 5 =q Inverted → 1 Injector and 4 producers Performance defined by q=f [ p W 1 ( t ) −p W 2 ( t ) ] q 1 =q q →q 2 =q 3 =q 4 =q 5 = q 4 r w 1 =r w 2 =r w 3 =r w 4 =r w 5 =r w d 2,3 = √ 2 d 2 = √ 2 d Drawdown In relation to well 1: 2 πkh μ [ p i −p W 1 ( t ) ] =−q [ p D ( 1 ,t D 1 ) +S 1] + q 2 p D ( d r w 1 ,t D 1 ) + q 3 p D ( d r w 1 ,t D 1 ) +q 4 p D ( d r w 1 ,t D 1 ) +q 5 p D ( d r w 1 ,t D 1 ) In relation to well 2: 2 πkh μ [ p i −p W 2 ( t ) ] =+q 2 [ p D ( 1 ,t D 2 ) + S 2 ] −q 1 p D ( d r w 2 ,t D 2 ) +q 3 p D ( √ 2 d r w 2 ,t D 2 ) +q 4 p D ( 2 d r w 2 ,t D 2 ) + q 5 p D ( √ 2 d r w 2 ,t D 2 ) Simplifying equation for well 1: 2 πkh μ [ p i −p W 1 ( t ) ] =−q [ p D ( 1 ,t D 1 ) +S 1] +4 q 4 p D ( d r w ,t D ) Simplifying equation for well 2: 2 πkh μ [ p i −p W 2 ( t ) ] = +q 4 [ p D ( 1 ,t D ) +S 2] −qp D ( d r w ,t D ) + 2 4 qp D ( √ 2 d r w ,t D ) + 1 4 qp D ( 2 d r w ,t D ) Combining equations for well 1 & 2 and dividing by q: 2 πkh μq [ p W 1 ( t)−p W 2 ( t) ] = 5 4 p D ( 1 ,t D ) + S 1 + 1 4 S 2 −2 p D ( d r w ,t D ) + 2 4 p D ( √ 2 d r w ,t D ) + 1 4 p D ( 2 d r w ,t D ) Rearranging for q:
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Transcript of Assignment 7
Assignment 7 – Mark Thomas CID: 00645751
Isolated inverted 5-spot pattern
Isolated →q1=q2+q3+q4+q5=q
Inverted → 1 Injector and 4 producers
Performance defined by q=f [ pW 1 (t )−pW 2 (t ) ]
q1=qq→q2=q3=q4=q5=q4
rw1=rw2=rw3=rw4=rw5=rw
d2,3=√2d2=√2d
Drawdown
In relation to well 1:
2πkhμ
[p i−pW 1 (t ) ]=−q [ pD (1 ,t D1 )+S1 ]+q2 pD( drw1 ,t D1)+q3 pD(drw1,t D1)+q4 pD( drw1 , tD1)+q5 pD(
drw1,tD 1)
In relation to well 2:
2πkhμ
[p i−pW 2 ( t ) ]=+q2 [ pD (1 ,tD 2 )+S2 ]−q1 pD ( drw2 , tD2)+q3 pD (√2drw2
,tD 2)+q4 pD( 2drw2 , tD2)+q5 pD (√2drw2
,tD 2)
Simplifying equation for well 1:
2πkhμ
[p i−pW 1 (t ) ]=−q [ pD (1 ,t D1 )+S1 ]+4q4pD ( drw , tD)
Simplifying equation for well 2:
2πkhμ
[p i−pW 2 ( t ) ]=+q4 [ pD (1 ,t D )+S2 ]−q pD ( drw , tD)+ 24 q pD(√2drw , tD)+ 14 q pD (2drw ,t D)
Combining equations for well 1 & 2 and dividing by q:
2πkhμq
[pW 1(t)−pW 2 ( t ) ]=54pD (1 , tD )+S1+
14S2−2 pD( drw , tD)+ 24 pD(√2drw ,t D)+ 14 pD( 2drw , tD)
Rearranging for q:
q=2πkhμ
[ pW 1(t )−pW 2 ( t ) ]54pD (1, tD )−2 pD ( drw , tD)+ 24 q pD(√2drw ,t D) +1
4pD( 2drw , tD)+S1+ 14 S2
Semi log approximations for line source and at finite distance:
pD (rD ,tD )=12 [ ln( tDrD2 )+0.80907]rD>1 pD (rD ,tD )=1
2[ ln(tD)+0.80907 ]rD=1
Substituting semi-logs into equation:
q=4 πkhμ
[ pW 1(t)−pW 2 (t ) ]54
[ ln (tD)+0.80907 ]−2¿¿
Rearranging:
q=4 πkhμ
[ pW 1(t)−pW 2 ( t ) ]54
[ ln (tD)+0.80907 ]−2[ ln (tD )+0.80907−2 ln ( drw
)]+ 24 [l n (tD )+0.80907−2 ln( drw )−ln (2)]+14 [ ln (tD )+0.80907−2 ln( drw )−2 l n (2)]+2(S¿¿1+ 14S2)¿
Cancelling the tD terms:
q=4 πkhμ
[ pW 1−pW 2 ]
−2[−2 ln ( drw )]+ 24 [−2 ln( drw )−ln (2)]+14 [−2 ln( drw )−2 ln (2)]+2(S¿¿1+ 14 S2)¿¿ 4 πkhμ
[ pW 1−pW 2 ]
4 ln( drw )−32 ln( drw )−ln (2)+2(S¿¿1+ 14S2)¿
¿ 2πkhμ
[ pW 1−pW 2 ]54ln( drw )−12 ln (2)+S1+ 14 S2
¿ 2πkhμ
[ pW 1−pW 2 ]
ln [ 1√2 ( drw )54 ]+S1+ 14 S2
Assuming that S1 = S2 = S
Equation for isolated inverted 5 spot pattern:
q=2πkhμ
[ pW 1−pW 2 ]
ln [ 1√2 ( drw )54 ]+ ln( 1e−S 54 )
=2 πkhμ
[ pW 1−pW 2 ]
ln [ 1√2 ( drwe )54 ]
Equation or isolated inverted n-spot pattern:
q=2πkhμ
[ pW 1−pW 2 ]
ln [ 1n−1√n−1 ( drwe )
nn−1 ]
