KAPPA DDA Book v5.12

25 / 743

Dynamic Data Analysis – v5.12.01 - © KAPPA 1988-2017 Chapte - p25/743
We now consider the case of a homogeneous, infinite reservoir produced by a vertical well with
constant wellbore storage and constant skin.
2.D.3 Derivation
We now have a slightly more complex problem to solve:
Homogeneous radial diffusion: 



















r
p
r
r rc
k
t
p
t
1
0002637 .0

Finite radius well:
kh
q
r
p
r sf
t r
w

2. 141
,
  




Initial pressure & infinite reservoir:   i p r tp   ,0     i r p trp 
 , lim
Wellbore storage & skin:
t
p
C qB qwf
sf 

 24 S
kh
q
p p sf
wf

2. 141  
We will now repeat the definition of the dimensionless terms and add dimensionless sandface
rate and dimensionless wellbore storage. The Skin, being dimensionless, does not need any
conversion:
Dimensionless radius, time & pressure:
w
D r
r
r 
2
0002637 .0
wt
D rc
kt
t
 
  
p p
qB
kh
p i D  
 2. 141
Dimensionless sandface rate & storage:
qB
q
q sf
D  2
8936 .0
wt
D rch
C
C


We now get the equivalent dimensionless problem:
Homogeneous radial diffusion: 


















D
D
D
D D D
D
r
p
r
r r t
p 1
Initial pressure & infinite reservoir:   0 ,0  
D DD r t p     0 , lim 
D DD t rp
Finite radius well: 1
,1
 







 D D t r D
D
D r
p
r
Wellbore storage & skin: D
t r D
D
D q
r
p
r
D D
 







 ,1
Sq p p
D D wfD  
D
wfD
D D dt
dp
C q  1
The solution process will be the same as the line source problem and will not be detailed here.
For the finite radius problem (without storage, without skin), the general form is like the line
source:
Generic Modified Bessel solution:   ) ( )( ) ( )( , 0 0 u r IuB u rKuA rup
D D D D  