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Файл:Подземная гидромеханика углеводородов. Ч. II. Математические 1D-модели многофазной фильтрации и процессов повышения нефтеотдачи. Учебное пособие
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7 3 *+)% [17]. (%$6 -+8;, && *$&"/$ / !'). 88, 30 +.;%
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5' &$)%6 *!$')1$ .<'4 3 $+8 ! ',) r, ' )$)%3+.;% ,!3//'
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r
h
Р и с. 88. >+4/% &!,2$3$2$ *+)%
3!4. ∆t 3 (+4/% ∆V 3$6 =% $#?=4 7
.,,
ttrcqttrqQ ∆⋅⋅=∆⋅=
*31
(%$ 5 3!4. '" (+4/% 9!" *$*!9/$ )9/' ) &$$! '/%$6 r+∆r 306 % $#?=4 7
*32
(313)
.,,
ttrrcqttrrqQ ∆⋅∆+⋅=∆⋅∆+=
(314)
4$4/% 3!4/' t 3 (+4/% $#?=4 *+)% ∆V )$ !5+$)8
7 3 &$+'9)%3
.,,2
3
202
π
+⋅∆⋅⋅⋅⋅=
tratrcrrhmQ
(315)
3!4. ∆t &$+'9)%3$ 7 '"4/'%). ' )%/% !3/04
()(
)
[
]
~
~
r
~
()(
)
[
]
=∆⋅∆+
−
()()()(
)
[
]
~~~
~
∂
∂
~
=
→
∂
∂
.,,2
4
π
ttrattrcrrhmQ ∆++∆+⋅∆⋅⋅⋅⋅=
(316)
)8
&$/-/%!-'. 7 !3/ )! /4, "/9/'; &$/-/%!-'' 3 (+-
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Q1 – Q2 = Q3 – Q4 (317)
ttrrctrcq ,,
.
ttrattrctratrcrrhm
π
m
[ ]
⋅
( ) ( )
∂
t
,, =
++
tratrc
q
2
π
⋅⋅
c
⋅
∂
r
rh
∆+−∆+−+∆⋅⋅⋅⋅= ,,,,2
.0
(319)
(318)
.
0
+ ,% "4%'%8, 9%$ *!' ∆r→0
,9=%$4 %$2$, 9%$ a(c) = α·c,
c
+
∂
t
>%$ ' )%8 '!/-'+8/$ ,!3//' #+/) &$/-/%!-''
7 3 3$ $/)0<//$4 *+)%.
+. !:/'. ,!3//'. /$#1$ '4$ " %8 /9+8/0 ' 2!/'9/0 ,)+$3'., &$%$!0 $!4,+'!,;%). )+ ,;<'4 $#!"$4.
/9+8/06 4$4/% 3!4/' t = 0 3 *+)% $%),%)%3,% 7, %$ )%8
c(r,0) = 0. (321)
9'/. ) 4$4/% 3!4/' t = 0 3 *+)% "&9'3%). 3$ /06
!)%3$! 7 ) &$/-/%!-'6 c = c0. &'4 $#!"$4, 2!/'9/$
q
( )
rr
⋅+⋅⋅⋅
απ
c
⋅
∂
r
rhm
.012=
(320)
,)+$3' '4% 3'
c(r),t) = с0. (322)
203
*$4$<8; "4/0 *!4//01
⋅
ξ
∂
∂∂∂
∂
τ
∂
∂∂∂
∂
∂
∂
(
)
2
ξ
=
2
−
c
,
2
2
rrrr−
c
&
τ
=
( )
1 rrhm
tq
2
2
( )
πα
−⋅⋅⋅+⋅
&
(323)
c
*!'3$ .% ,)+$3'. " 9' (320)–(322) & #$+ , $#/$4, +. !:/'.
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2
=
∂
c
=
ξ
∂
r
=
t
π
c
∂
r
c
c
=
∂
t
τ
r
2
( )
−
rr
&
q
2
( )
&
c
2
c
;
(324)
ξ
∂
c
2
−⋅⋅⋅
rrhm
c
.
(325)
τ
∂
7$ )%/$3&$6 *$+,9//01 )$$%/$:/'6 3 ,!3//' (320) 4$5/$ *$+,9'%8 30!5/'
cc
+
∂
∂
ξτ
.0=
(326)
9+8/0 ' 2!/'9/0 ,)+$3'., )$$%3%)%3//$, *!'/'4;% 3'
с
ξ
( )
0,
=
,
τ
=
,00
0
(327)
.
cc
",+8%%$4 !:/'. " 9' (326)–(327) .3+.%). )')%4 "3')'4$)%6
0
( )
c
с
τξ
=
,
,0
≤
>
,,
τξ
(328)
.
τξ
&'4 $#!"$4, *$+$5/' !$/% 7 $*! +.;% ) *$4$<8;
,!3//'. ξ = τ.
7!1$ . & !"4!/04 3+'9'/4, *$+,9;%
⋅
hm
⋅
tq
( )
hm
tq
( )
+⋅⋅⋅
1
+⋅⋅⋅
1
απ
,
απ
(329)
.
( )
,
trc
=
2
,0
2
+>
rr
c
≤
2
20
,
+
rrc
c
204
%); )+ ,%, 9%$ %&,< *$+$5/' !$/% 7 4$5/$
(
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(
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t
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⋅
tq
( )
hm
.
(330)
απ
+⋅⋅⋅
1
( )
2
+=
rtr
c
'!/-'!,. $# 9)%' ,!3//'. (329) *$ t, /1$ .% )&$!$)%8
*!$ 3'5/'. !$/% 7:
( )
tv
dt
tdr
==
( ) ( )
12 ⋅⋅⋅+⋅
q
.
(331)
πα
trhm
93' /$, 9%$ )&$!$)%8 *!$ 3'5/'. !$/% 7 3 )+,9 ! '+8/$6 '+8%!-'' * % ) %9/'4 3!4/' *$ "&$/,, 4%4%'9)&. "*')8 &$%$!$2$ '4% 3' (331).
)+' /$#1$ '4$ !))9'%%8 )&$!$)%8 *!$ 3'5/'. !$/% 7
3 /9+8/06 4$4/% 3!4/' t = 0, %$ 3 ,!3//' (331) )+ ,% *$ )%3'%8 r(t0) = r).
+. !)9=% )&$!$)%' !$/% 3 &$/9/06 4$4/% 3!4/' t* 3
,!3//' (331) /,5/$ *$ )%3'%8 r(t*) = r&, %& && !$/% 7 *$ $6 =% & $#03;<'4 )&35'/4.
!4. t* *$ 1$ !$/% 7 & $#03;<'4 )&35'/4 4$5/$
!))9'%%8 ) *$4$<8; )$$%/$:/'. (330), 3 &$%$!$ +. (%$2$ *$ )%3+.;% "/9/' r(t) = r&. 7$)+ *!$#!"$3/'6 *$+,9;%
11
hm
t ⋅
=
+⋅⋅⋅
&
q
7,)%8 3 4$4/% 3!4/' t =
2
2
rr
≈−⋅
c
"&/9'3%). )$" /' $%$!$9&'
hm
+⋅⋅⋅
απαπ
q
2
.*
r
&
(332)
' /9'/%). *!$%+&'3/' = *$ *+)%, 3$ $6, / )$ !5<6 7.
+. $*! +/'. 3!4/' $!4'!$3/'. $%$!$9&' /$#1$ '4$ !))9'%%8 )&$!$)%8 *!4</'. %0+ $%$!$9&' *$ *+)%,. (%$6 -+8; 303$ .% ,!3//', $*')03;< !)*! +/' &$/-/%!-''
&%'3/$6 *!'4)' / )% '' *!$%+&'3/'. $%$!$9&' 7 "&9'34$6 3$ $6 [17].
/$38 30 +.;% (+4/% $#?=4 *+)% ∆V = 2πFh·rF∆r ' !))4%!'3;% #+/) $#?=4 7, !)),5 . )+ ,;<'4 $#!"$4.
205
3!4. ∆t 3 (+4/% ∆V 3$6 =% $#?=4 7
(
)
(
)
()(
)
[
]
~~~
()(
)
[
]
~~~
()(
)
[
]
()()()(
)
[
]
~
~,~,~
~,~
∆+−∆+−+∆⋅⋅⋅⋅=
π
∂
∂
(
)
~
∂
∂
*
t
(
)
(
)
.,
ttrсqQ ∆⋅⋅=
1
(%$ 5 3!4. '" (+4/% ∆V 306 % $#?=4 7
2
4$4/% 3!4/' t 3 (+4/% $#?=4 *+)% ∆V )$ !5+$)8
7 3 &$+'9)%3
3
3!4. ∆t &$+'9)%3$ 7 '"4/'%). ' )%/% !3/04
4
$)%3+.. ,!3//' #+/), "*')03;%
$%&,
π
π
Q1 – Q2 = Q3 – Q4, (337)
(333)
.,
ttrrcqQ ∆⋅∆+⋅=
(334)
.,,2
+⋅∆⋅⋅⋅⋅=
tratrcrrhmQ
(335)
.,,2
ttrattrcrrhmQ ∆++∆+⋅∆⋅⋅⋅⋅=
(336)
=∆⋅∆⋅∆+− tttrrctrcq ,,
2 ttrattrctratrcrrhm
+.% $# 9)%' *$+,9//$2$ ,!3//'. / ∆VF∆t ', ,)%!4+.. ∆r
' ∆t & /,+;, *$+,9;%
[ ]
⋅
m
)*$+8",. 30!5/' +. $*! +/'.
!"$3/'. ' *$+,9;% ,!3//' !)*! +/'. &$/-/%!-'' 7 3
*+)% / )% '' *!$%+&'3/'. $%$!$9&' 3$ $6 3 3'
!&%!/$, 9%$ 3 4$4/% 3!4/' t =
( ) ( )
∂
t
c
+
∂
t
~
,, =
++
tratrc
q
( )
12
q
2
π
~
⋅+⋅⋅⋅
rhm
απ
c
.
⋅
∂
⋅⋅
rh
c
⋅
∂
r
3$ 3)1 )9/'.1 *+)%,
0
r
cа
=
(339)
, *!$'"3$ .% *!$#-
.0
(340)
.,
(338)
9!" &$%$!0 *!$:+ !$/% $%$!$9&' 7, &$/-/%!-'. 7 !3/
&$/-/%!-'' "&9&'. &'4 $#!"$4, /9+8/$ ,)+$3' '4% 3'
.,
0,
=≤= trrctrc
0
206
**
(341)
9'/. ) 4$4/% 3!4/' t =
*
t
*
t
(
)
(
)
(
)
$%$!$9& #, % *!$%+&'3%8).
3$ $6, / )$ !5<6 7. 7$(%$4, 2!/'9/$ ,)+$3' *!'4% 3'
c(r),t) = 0, t >
(342)
.
/+$2'9/$ *!$- ,!, $*')//$6 30:, 33$ .% #"!"4!/0
*!4//0 ξ ' τ, *$%$4 3/$38 *!1$ .% & !"4!/04 3+'9'/4,
*$+,9. *!' (%$4
−⋅
ttq
*
( )
+⋅⋅⋅
1
hm
( )
−⋅
ttq
*
( )
1
hm
,
~
απ
(343)
.
~
+⋅⋅⋅
απ
( )
,
trc
=
2
,0
0
,
2
%
2
2
+<<
rrr
c
2
2
+≥≥
rrrc
c
&'4 $#!"$4, %&,< *$+$5/' %0+ $%$!$9&' 7 / )% '' *!$%+&'3/'. = 3$ $6 4$5/$ $*! +'%8 *$ )$$%/$:/';
−⋅
( )
1
hm
ttq
*
+⋅⋅⋅
.
~
(344)
απ
( )
2
+=
rtr
c%
'!/-'!,. $# 9)%' ,!3//'. (344) *$ t, /1$ .% )&$!$)%8
*!$ 3'5/'. %0+ $%$!$9&' 7:
%
( )
tv
%
dt
tdr
==
( )
12 ⋅⋅⋅+⋅
~
q
.
(345)
( )
πα
trhm
%
0+ $%$!$9&' *!'1$ '% 3 3'5/' 3 4$4/% "3!:/'.
$!4'!$3/'..
+ $3%+8/$, /9+8/. )&$!$)%8 3'5/'. )$)%3'%
q
%
( )
tv
*
=
~
( )
12 rhm
πα
.
(346)
⋅⋅⋅+⋅
)
&$/9/06 4$4/% 3!4/' t*, &$2 !$/% ' %0+ $%$!$9&' 7
$ /$3!4//$ *$ $6 ,% & $#03;<'4 )&35'/4, r%(t*) = rк. &$!$)%8 %0+ $%$!$9&', )$$%3%)%3//$, #, % !3/
q
%
( )
*
tv
=
~
( )
12 rhm
πα
.
⋅⋅⋅+⋅
(347)
&
207
!3/'3. )&$!$)%' 3'5/'. %0+ ' !$/% $%$!$9&' 7,
*
t
(
)
*
t
*
t
*
t
(
)
*
t
[
]
4$5/$ "4%'%8, 9%$ %0+8/. 9)%8 $%$!$9&' 3'5%). ) $*!5/'4. >%$ "/9'%, 9%$ $#?=4 $%$!$9&' )$ 3!4/4 ,4/8:%)..
))9'%03;% 3!4. $!4'!$3/'. $%$!$9&'
+%8, !,&$3$ )%3,.)8 )+ ,;<'4' )$$#!5/'.4'.
" $*! +/'. $*%'4+8/$2$ $#?=4 $%$!$9&' )+ ,%, 9%$ =
!$/% ' %0+ $+5/0 $ /$3!4//$ $)%'2/,%8 $#03;<6 2+!'
3 4$4/% 3!4/' t*. 7$+$5/' !$/% $%$!$9&' 7 3 4$4/%
3!4/' t* $*! +'%). '" )$$%/$:/'.
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tq
( )
*
7$+$5/' %0+ $%$!$9&' 7 3 (%$% 5 4$4/% 3!4/' $*! +'%). '" )$$%/$:/'.
( )
*
2
rtr =
+=
c
2
rtr =
+=
c%
⋅
( )
1
hm
*
−⋅
( )
hm
r
+⋅⋅⋅
απ
ttq
*
~
1
+⋅⋅⋅
απ
r
. >%$ 4$5/$ ) -
.
&
(348)
.
&
(349)
2
(349) 30!5;% 3!4.
(348) – /+$2'9/04 $#!"$4 3!4. t*. '%$2 *$+,9;%
!'%+8/$ $*! +.;% &$$! '/%, !$/% $%$!$9&' *$ "3!:/'' =
$!4'!$3/'. r(
– 3!4. $!4'!$3/'. $%$!$9&'.
>%' 3 ,!3//'. !:;% $%/$)'%+8/$
9!" !))%$./'. r& ' r), '" )$$%/$:/'.
22
t
=
*
))9'%03;% /9+8/06 $#?=4 $%$!$9&'. (%$6 -+8; *! 3-
):
( )
*
#?=4 $%$!$9&' )$)%3'%
cк
2
+=
rtr
c
hmrr
q
⋅
tq
hm
~
( )
ααπ
−⋅⋅⋅⋅−
*
( )
απ
+⋅⋅⋅
1
.
. " )$$%/$:/'.
.
(350)
(351)
2
( )
*
$%
208
π
⋅
2
hrtrmV
c
tq
=⋅−⋅⋅=
*
.
(352)
α
+
1
7!'/.%$ 30!5%8 $#?=4 $%$!$9&' &%'3/$6 *!'4)' 3 *!$-/-
(
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⋅
(
)
(
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(
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%1 *$!$3$2$ $#?=4 *+)%.
+'9'/, *$!$3$2$ $#?=4 (+4/% *+)% 4$5/$ *! )%3'%8 3
)+ ,;<4 3' :
2
rr
c
2
&
tq
*
−⋅−
2
rr
)
2
( )
~
αα
+⋅
1
( )
r
&
2
π
$%);
V
$%
V
*$!
$+'9)%3$ 7, /$#1$ '4$ +. )$" /'. $%$!$9&', $*! +'%). '" )$$%/$:/'.
3$ $/)0<//06 ,9)%$& *+)% :'!'/$6 b = 300 4, %$+<'/$6
h = 15 4, *$!')%$)%8; m = 0,25 ' ) !))%$./'4 45 , /2/%%+8/$6 ' $#03;<6 2+!.4' l = 500 4 9!" /2/%%+8/,; 2+-
=
⋅
tq
*
+
1
α
2
&*$!
2
&
⋅
Ситуация 6.1
hrrmV ⋅
c
2
rr
)
V
=⋅−⋅⋅=
( )
−⋅−
2
⋅⋅
rtq
*
&
*3 = V$%
2
( )
&
=
· )0. (355)
2
,
r
&
~~
( )
−⋅−
αααα
α
(353)
.
(354)
!; "&9'3%). 3$ /06 !)%3$! 7 ) &$/-/%!-'6 с0 = 0,001 '
%4*$4 "&9&' q = 400 43/),%. %$!$9& *!$%+&'3%). *$ *+)%,
3$ $6 ) %4 5 !)1$ $4 q.
)% '' $!4'!$3/'. $%$!$9&' 7 )$!#'!,%). *$3!1/$)%8; *$!$ 0 *$ "&$/, /!', $!4,+ &$%$!$2$ '4% 3'
a(c) = α·c, *!' (%$4 &$('-'/% )$!#-'' α = 0,3.
)% '' *!$%+&'3/'. $%$!$9&' 3$ $6 *!$')1$ '% )$!#-'.
7 *$ "&$/, (312), *!'9=4 &$('-'/% )$!#-''
!#,%).:
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