PRINCIPLE OF SUPERPOSITION
The The princ rincip iple le state tatess that that:: addin dding g soluti lution onss to a linear differential equation results in a new solution to that differential equation, but for for differ ferent boundary conditions. conditions. SUPERPOSITION IN SPACE SPACE
Consider the two-well infinite system sketched in Fig. To estimate the pressure change at the observation point N (or shut-in wells caused by well ! and well "# we can write: Point N r 2
90° r 1
Well 1, q
1
Well 2, q2 ∆ P = ∆ P on N caused by well ! + ∆ P on N caused by well N
$t any point# $t a well# Thus:
∆ P =
∆ P =
!%!."qµ B
!%!."qµ B kh
kh
P D (r D # t D
[ P D (r D # t D + s ]
&nternal !
"
= ∆ P + ∆P N #! N # "
∆ P N =
!%!."µ ( qB ) P D (r D N #! # t D + ( qB ) P D (r D N # " # t D kh
[
]
'tending to a large number of wells# the pressure change at any point is: ∆ P ( r # t ) =
!%!." µ kh
n
∑ qi Bi P D (r Di # t D i =!
$t the well ∆ P ( r w! # t =
!%!." µ kh
n
∑ qi Bi P D (r Di # t D + i =!
!%!." µ qB kh
s
EXAMPLE &f point N in Figure ! is a well producing an oil rate of !)* +,# compute its flowing pressure at ** hours urs. The foll follo owing inform ormatio tion is available for the reservoir and the wells:
&nternal "
Point N r
2
90° r
1
Well 1, q
1
Well 2, q2 Figure: !. /uperposition in space ,i 0 "12) psia h 0 )* ft k 0 !" md φ 0 "* 3 µ 0 ) cp ct 0 )!*-) 4psi + 0 !. bbl4/T+ r ! 0 )** ft r " 0 ))* ft 5! 0 !** /T+4 5" 0 !2* +,
s well at point N 0 ! r w well at point N 0 *. ft s well " 0 " well ! 0 *.% ft r w well " 0 *.% ft s well ! 0 -! SOLUTION
rw
/uperposition e5uation applied to this eample is:
∆ P N =
!%!." µ q N B
kh !%!." µ q" B kh
[ P (r D
D 6 r = *. ft
# t D + s N ] +
!%!." µ q! B kh
P D (r D 6 r = )** ft # t D +
P D (r D 6 r = ))* ft # t D
Notice that only pressure drop dro p due to well damage takes place in well at point N.
&nternal
∆ P N =
1*.7()(!)*(!. [ Ei (− x r = *. ft + " s N ] + 1*.7()(!**(!. Ei(− x r = )** ft (!"()* (!"()*
+
1*.7()(!2*(!. Ei ( − x r = ))* ft (!"()*
xwell N
=
xwell N −!
=
xwell N − " ∆ P N =
8%2(*."()(!.) × !* −) (*. " !"( "%
8%2(*."()(!.) × !*−) ()** " !"( "%
=
2.71 x!* − 2
=
*."%*8
8%2(*."()(!.) × !* −) ())* "
= *."8!% !"("% 1*.7()(!)*(!. 1*.7()(!**(!. 1*.7()(!2*(!. *.2)7 + *.1"" [!).87 + "] + (!"()* (!"()* (!"()* =
∆ P N = [!1"." + "".2)] + 7.2) + 8.787 = "!*.2 psia
,wf
0 ,i 9 ,N
Flowing Flowing pressure 0 "12)-"!*.2 0 ")1%." psia
&nternal %
SUPERPOSITION IN TIME
/uperposition in time applies for single-well systems with varying flow rates. Consider a single well with a production rate schedule schedule as shown shown in Fig. Fig. eq t 2 a R w o l F
q2 - q1
q
1
t1 Producing Time
Figure: ariable rate 9 /uperposition /uperposit ion in time Well 2
Well 1 s=-1
s=2
r w= 0.5 ft
r w= 0.5 ft 250 ft
70
120
60
q
q 20
15
10
t
t
Figure !. 'ample of superposition in time
&nternal )
To perfor perform m superp superposi ositio tion n calcu calculat lation ionss the singl singleewell may be visualized as two wells located at the same point# with one producing at rate 5 ! from t 0 * to t and the second (imaginary well# producing at rate (5"-5! starting at time t# and continuing for a time period (t-t!. Thus# the pressure drop at the single well is: ∆ P =
!%!." µ B kh
[ q P (r ! D
D
!
# t D + ( q"
−
q! ) P D ( r D ! # ( t − t ! ) D + s ]
The general form of this e5uation is: ∆ P =
!%!." µ kh
n
∑ { (qBi − (qBi !}[ P D (r D # ( t − t i ! ) D + s] −
!
−
i =!
EXAMPLE
;efe ;eferr to the the info inform rmat atio ion n for for an infi infini nite te twotwo-w well ell reservoir system given in Figure !. $dditional relevant parameters are provided below:
&nternal 7
k 0 !** md ,i 0 ")** psi h 0 )* ft
φ 0 !* 3 µ 0 " cp
+ 0 !." bbl4/T+ ct 0 "!*-)4psi
a 'stimate 'stimate the pressure pressure in well ! after it has produced produced for !* hours. b
a The pressure drop in well ! after producing for !* hours is determined as follows: ∆ P !* hrs # well ! =
x well !
=
!%!." µ q!"* B kh
[ P (r D
D
]
= !# t D 6 t =!* hrs + s! +
8%2(*.!( "( " × !* −) (*.) " (!**(!*
=
8.%2 × !* −1
=
*."1
'i (-> 0 ?n (!.12! 'i 0 !."8 xwell "
=
8%2(*.!("(" × !*−) (")* " (!**(!*
From Table 'i 0 !.*27
&nternal 1
!%!."µ q"* B kh
[ P (r D
D 6 r =")* ft
# t D 6 t =!* hrs ]
The pressure drop rop at well ! after !* hours of production is: ∆ P !* hrs # r
D =!
1*.7( "(!." [ (!"*(!."8 − ! + "*(!.*27] !**()*
=
∆ P !* hrs # r D =! = ()%.*) − 2.! + *.17 = %7.7) psi
,wf 0 0 ,i - d, 0 ")** - %7.7) 0"%).% psi
bTo b To estimate estimate the pressure drop in well " after !) hours# consider two flow rates in well "# as follows: ∆ p!) hrs# well " =
xwell !
=
!%!." µ q!"* B
[ P (r D
kh
D 6 r =")* ft
8%2(*.!("(" × !*−) (")* " (!**(!)
# t D 6 t =!) hrs ] +
=
*.!)2
=
7." × !*− 1
!%!." µ q1*−"* B kh
[ P (r D
D
= !# t D 6 t =!)−!* hrs
'i well !0 !.%" xwell "
=
8%2(*.!( "(" × !* −) (*.) " (!**(!) − !*
'i well "0 !.781 The pressure drop rop at well " after !) hours of production is: ∆ p!) hrs # well " = ∆ p!) hrs # well " =
1*.7( "(!." !**()*
[ (!"*(!.%" + )*(!.781 + "]
).11 + [ ".!7 + .28]
=
"." psi
,wf 0 ,i 9 d, 0 ")**-"." 0 "%71.72 psi &nternal 2
PROBLEMS (Home Work) Work) !. $n oil well is producing producing at a constant flow flow rate of ")* /T+4 from a very large reservoir. $n observation shut-in well is located loca ted ** * * ft away. away. ;ock and fluid properties are listed below. 'stimate the sandface pressure at both the producing well and the observation well at the end of two days of production. The initial reservoir pressure is 7** psia. @ther @ther relevant data data are:
+ 0 !." rb4/T+ φ 0 !% 3 ct 0 !7!*-7 4p 4psi
rw 0 in k 0 2* md t 0 %2 hrs
µ 0 !." cp h 0 !! ft
*". $ well is producing at a rate of %** /T+4 from a reservoir that has the following rock and fluid characteristics: µ 0 cp + 0 !.") rb4/T+ rw 0 7 in φ 0 * 3 k 0 )* md h 0 * ft co 0 2!*-7 4p 4psi cf 0 0 "!*-7 4psi $fter what value of the flowing time is the appr appro oiimatio tion 'i(- i(- ln(!.12 > valid for this well4reservoir= a
&nternal 8
*. Two oil wells are producing at constant flow rates of 5! 0 ")* +, and 5 " 0 ** +,# respectively# from a very large reservoir.
rw 0 *. ft k 0 %* md t 0 " days
µ 0 !.! cp h 0 %* ft ,i 0 7** psi
'sti 'stim mate ate the the sand sandfa face ce pres pressu sure re at the the prod produc ucin ing g wells and the pressure drop at the observation well (shut-in well at the end of two days of production.
Observation well
300 ft
Well 2
Well 1 400 ft
Fig.
&nternal !*
located )** ft away from the producing well. ;ock and fluid properties are listed below. + 0 !." rb4/T+ φ 0 !7 3 ct 0 !2!*-7 4p 4psi
µ 0 *.%% cp rw 0 in k 0 ") md h 0 % ft ,i 0 ")** ft
'stimate the pressure drop at the observation well when the producing well has been shut-in for one day# after the five days of production.
NOMENCLATURE
&nternal !!
+ +g b by c C C$ Cf d 'i k h m N , ,A
oil vo volumetric ric fa factor# rb4/T+ gas volumetric factor# bbl4/CF shortest distance in the -direction from well to boundary shortest distance in the y-direction from well to boundary
compressibility# !4psi wellbore st storage# bb bbl4psi shape factor# dimensionless dimensionless fracture conductivity distance between wells# ft eponential-integral function reservoir permeability# md reservoir thickness# ft semilog slope reference point point pressure# psi pres presssure deriv erivat ativ ivee# ps psi4h i4hr $verage pressure# psi ,wf well flowing well pressure# psi , dimensionless pressure ,A dimensionless pressure derivative ,int intercepting pressure at Bero time (Cartesian plot# psi 5 @il flow rate# +, 5g gas flow rate# /CF4 r radius# ft r dimensionless radius r ; ; dimensionless distance of the real well r & dimensionless distance of the image well & r w wellbore radius# ft s skin factor P
&nternal !"
T t t f D z
reservoir temperature# ; time# hrs dimensionless time r "4%t half-fracture length# ft by4b $verage gas compressibility factor
GREEK SYMBOLS
∆ change# drop ∆, pres pressu sure re dif diffferen erence ce## psi psi ∆,c pressure correction at the beginning of the test# psi
&nternal !
∆,A Change of rate of pressure with time (pressure derivative# psi
∆tc φ ρ µ
time correction at the beginning of the test# psi ,orosity ensity iscosity# cp
SUBSCRIPTS
* f g o w t
reference dimensionless formation gas oil wellbore# well total
&nternal !%
