Well Testing
2
Diffusivit y equation, Eq.(A.9),
is derived based on several important assumption s, such as
(1) The single - phase liquid flowing has small and constant compressib ility
(2)  is independen t of pressure
(3) k is constant and the same in all direction (isotropic )
(4)  is constant
(5) pressure gradients and c are small
(6) one - D flow
3
1   p 
c
p
r  
r r  r  2.634 10  4 k t
1   p  1 p r
c
p
 r  

r r  r  r r r 2.634 10  4 k t
 2 p 1 p
c
p


     (1.1)
2
4
r
r r 2.634 10 k t
This equation is called the diffusivit y equation
2.634 10  4 k
 The hydraulic diffusivit y  
c
Units of equation (1.1)
p [] psi
 []cp
r [] ft
 [] fraction
c []vol / vol / psi[]vol /( vol  psi )[ ]1 / psi
k []md
t []hr
 [] ft 2 / hr
4
Dimensionl ess analysis
 2 p 1 p
c
p


2
r r 2.634  10 4 k t
r
p  f q, r , rw ,  , c, , k , t , h 
5
6
7
Dimensionl ess form of diffusivit y equation of Eq.(1.1) - - - - (oil reservoir)
 2 p 1 p
c
p


     (1.1)
2
4
r r 2.634  10 k t
r
2

p
1 p
c
p
rw2 2  rw2
 rw2
r r
r
2.634  10 4 k t
2 p
1
p
p


2
r  r 
 2.634  10 4 kt 
r 
    

 
2
crw
 rw   rw 


 rw 
 2 p 1 p
p


rD2 rD rD t D
 kh( pi  p )  1 
 141.2qB   r r

 D D
 2 p D 1 p D p D


2
rD rD
t D
rD

rD2
where
r
rD 
rw
2.634  10 4 kt
tD 
crw2
 kh( pi  p ) 
  kh( pi  p ) 

 141.2qB  t  141.2qB 



D 
kh( pi  p )
where
pD 
141.2qB
Note : q[]STB / D ; B[]RB / STB ; qB[]RB / D ; qB  const . rate
8
For compressib le fluid (for z  const.)
From Eq.(A.5) such as
1   rk p 
1
  

 

( A.5)
4
r r   r  2.634  10 k t
for
k  const .   const .   const.
1 k   p 


r




r  r  r  2.634  10 4 t
1   p 

 p
or
r


     ( A.5a)


4
r r  r  2.634  10 k p t
From Equation of state
pV  nzRT
m
m
p( )  ( ) zRT

M
MP
M p

(
)      ( A.5b)
zRT
RT z
9
Substituti ng Eq.(A.5b) into Eq.(A.5a), we have
1   M p p 

  M p  p
r



r r  RT z r  2.634 10  4 k p  RT z  t
1   p p 

  p  p
r



 
r r  z r  2.634 10  4 k p  z  t
Ideal gas, z  1 or z  const .
1   p 

p p
 rp  
r r  r  2.634 10  4 k p t
1   1 p 2 

p
 r
 
r r  2 r  2.634 10  4 k t
1 1   p 2 

1 p 2
 r
 
2 r r  r  2.634 10  4 k 2 p t
 p 2
p
p
1 p 2 


 2p


t
t 2 p t 
 t
1   p 2 

1 p 2
r

r r  r  2.634 10  4 k p t
1  2 p 2 1 p 2

p 2
c 
r


r r 2 r r 2.634 10  4 k
t


1
 c 
for gas 
p


 2 p 2 1 p 2
c
p 2


     ( A.5c)
r 2 r r 2.634 10  4 k t
10
Dimensionl ess form of diffusivit y equation of Eq(A.5c)
 2 p 2 1 p 2
c
p 2


     ( A.5c)
2
4
r
r r 2.634 10 k t
2 2
2
c
p 2
2  p
2 1 p
2
rw
 rw
 rw
2
r
r r
2.634  10  4 k t
2 p2
1
p 2
p 2


2
4
 r   r   r   2.634 10 kt 
     
2


r
r

c

r
r
w
w
w






 w
 2 p 2 1 p 2 p 2


2
rD
rD rD t D
where
r
rD 
rw
2.634 10  4 kt
tD 
crw2
2
2
 kh( p i2  p 2 )  1   kh( p i2  p 2 ) 
  kh( p i  p ) 






1422
q

zT
r

r
1422
q

zT

t
1422
q

zT

 D D 


D 

kh( p i2  p 2 )
 2 p D 1 p D p D


where
pD 
rD2
rD rD t D
1422qzT

rD2
11
For non - ideal gas
From Eq.(A.5) such as
1   rk p 
1
  

 

     ( A.5)
4
r r   r  2.634  10 k t
For k  const .   const .
1   r p 



 

     ( A.5d )
r r   r  2.634  10 4 k t
From Equation of state
pV  nzRT
m
m
p ( )  ( ) zRT

M
MP
M p

(
)      ( A.5b)
zRT
RT z
12
Substituti ng Eq.(A.5b) into Eq.(A.5d)
1   r pM p 

  pM 

 


r r   zRT r  2.634  10  4 k t  zRT 
1   p p 

  p
 r
 
 
r r  z r  2.634  10 4 k t  z 
p
dp
po z
2p
d 
dp
z
p
1
p p 1 
 dp  d 

z
2
z r 2 r
Define   2 
p
  p    p  p
   
t  z  p  z  t
 p  p
  cg 
 z  t
  p 2 p p


t
p t z t
p z 

t 2 p t
1 
1  p M 



 p p M p  z RT 
z RT
  p p
    cg
p  z  z
sin ce c g 
13
p z 

t 2 p t
  p   p  z  c g 
     cg 

t  z   z  2 p t
2 t
1   p p 

  p
 r
 
 
r r  z r  2.64 10  4 k t  z 
c g 
1   1  


r

r r  2 r  2.64 10  4 k 2 t
c g
1    


     (1.2a )
r

4
r r  r  2.64 10 k t
cg
 2 1 

 2 

r
r r 2.64 10  4 k t
 2 pD 1 pD pD



2
rD
rD rD t D
where
2.64 10  4 kt
tD 
c grw2
pD 
kh  pi    p 
1424qT
14
For simultaneo us flow of oil, gas and water
ct
1   p 
p
r

     (1.3)


4
r r  r  2.634  10 t t
where
ct  the total system compressib ility
 S o co  S w c w  S g c g  c f      (1.4)
t  the sum of the mobilities of the individual phases
 ko k g k w 
      (1.5)





 o g w 
15
From Eq.(1.2a) such as
1    
c

r

r r  r  2.64  10 4 k t
k 1    
c


r

 r r  r  2.64  10 4 t
For oil flow
co S o 



4
 2.64  10 t
For water flow
k o 1   
r
 o r r  r
k w 1    
c w S w 
r

 w r r  r  2.64  10 4 t
For gas flow
k g 1    
c g S g  
r

 g r r  r  2.64  10 4 t
Summation
 k o k w k g  1     co S o  c w S w  c g S g  




r

4




r

r

r
t
2
.
64

10


w
g 
 o
16
1.3 Solution to Diffusivity Equation
Equation of diffusivit y
 2 p 1 p
  c 
p

     (1.1)
2
4
r r 2.634  10 k t
r
• There are four solutions to Eq.(1.1) that are particularly useful
in well testing:
(1) The solution for a bounded cylindrical reservoir
(2) The solution for an infinite reservoir with a well considered to
be a line source with zero wellbore radius,
(3) The pseudo steady-state solution
(4) The solution that includes wellbore storage for a well in an
infinite reservoir
17
Equation of diffusivit y
 2 p 1 p
  c   p


     (1.1)
2
4
r
r r 2.634  10 k t
• The assumptions that were necessary to develop Eq.(1.1)
(1) Homogeneous and isotropic porous medium of uniform
thickness,
(2) Pressure-independent rock and fluid properties,
(3) Small pressure gradient,
(4) Radial flow
(5) Applicability of Darcy’s law ( sometimes called laminar flow )
(6) Negligible gravity force.
18
Constant rate, Infinite reservoir case
 2 p 1 p
  c 
p

r 2 r r 2.634 10  4 k t
 2 pD 1 pD pD


rD2
rD rD t D
kh( pi  p )
2.634 10  4 kt
where pD 
tD 
141.2qB
crw2
boundary and Initial conditions :
(1) p  pi at t  0 for all r
 qB
 p 
(2)  r 

for t  0

r
2


0
.
001127
kh

 r  rw
(3) p  pi as r   for all t
Dimensionl ess form
(1) pD  0 at t D  0 for all rD
 qB
 qB
 141.2qB
 p 
(2)  r 



kh
 r  r  rw 2  0.001127kh 0.00708kh
 p 
  rD D 
 1
 rD  rD 1
or
pD
 1
rD
(3) pD  0 at rD   for all t D
19
Infinite Cylindrica l Reservoir with Line - Source Well
Assume that
(1) a well produces at a constant rate qB ,
(2) the well has zero radius,
(3) the well drains an infinite area (i.e. p  pi that as r   )
(4) the reservoir is at uniform pressure, pi , before production begins.
20
Under thos e conditions , the solution t o Eq.(1.1) is
 2 p 1 p
  c 
p

     (1.1)
2
4
r
r r 2.634 10 k t
qB   948ct r 2 
      (1.7)
p  pi  70.6
Ei 
kh 
kt

 or
1  rD2 
      (1.7a )
pD t D , rD    Ei  
2  4t D 
 or
1  rD2 
      (1.7a )
pD   Ei  
2  4t D 
 where pD 
kh( pi  p )
0.000264kt
r
tD 
r

D
141.2qB
crw2
rw
Where p is pressure (psi) at distance r (feet) from the well at time t (hours), and
Ei  x    

x
e u
x2
x3
(1) n x n
du  ln( x)  0.5772  x 


u
2  2! 3  3!
n  n!
21
Eq.(1.7a) [or Eq.(1.7)] is called the exponentia l - integral solution
or the line - source solution
or the Theis solution
Ei ( x)  ln( x)  0.5772
for x  0.0025

1  rD2 
1   rD2 
   ln 
  0.5772
pD   Ei  
2  4t D 
2   4t D 

pD

1   tD 
 ln  2   0.80907
2   rD 

for x  0.0025
t
 D2  100      (1.7c)
rD
22
23
24
From Eq.(1.7a) such as
1  rD2 
      (1.7 a )
pD   Ei  
2  4t D 

1   tD 
 ln  2   0.80907       (1.7b)
2   rD 

Eq.(1.7b) may be use when
tD
 2  100      (1.7c)
rD
or
3.79 105 ctrw2
t
k
(rD  1)
25
But the difference between Eq.(1.7a) and Eq.(1.7b) is only about 2% when
tD
5
rD2
Thus, for practical purpose, the log approximat ion to the exponentia l
integral is satisfacto ry when th e exponentia l integral is satisfacto ry.
From Eq.(1.7b) such as
pD 

1   tD 


ln

0
.
80907


2   rD2 

At wellbor e , i.e.,
rD  1
1
ln t D   0.80907
2

kh( pi  pw ) 1   2.64 10  4 kt 


 ln 

0
.
80907

141.2qB
2   ct rw2 


1   2.64 10  4 kt 


 ln 
 ln( 2.2458)
2   ct rw2 

1   2.2458  2.64 10  4 kt 

 ln 
2
2 
ct rw

70.6qB   5.9289 10  4 kt 

 pi  pw 
ln 
kh  
ct rw2

70.6qB   1688ct rw2 
      (1.7 d )

ln 
kh  
kt

pD (t D ) 
26
From Eq.(1.7b) such as
rr
70.6qB   5.9289 10  4 kt 
 pi  p 
ln

kh  
ct r 2

2
70.6qB   1688ct r 

     (1.7d )
ln

kh  
kt

27
70.6qB
 pi  pw 
kh
  5.9289 10  4 kt 

ln 
2

c
r
t w

 
70.6qB   1688ct rw2 
      (1.7 d )

ln 
kh  
kt

70.6qB

kh
  1688ct rw2 





ln

ln
t
 


k




70.6qB
 pw  pi 
kh
  1688ct rw2 





ln

ln
t
 


k

 

70.6qB
 pw  pi 
kh
  1688ct rw2  70.6qB
 
ln t 
ln 
k
kh

 
Question:
Why does pw > pi for certain t ?
28
In practice, most wells have reduced permeabili ty (damage) near
the wellbore resulting from drilling or completion operations .
Many other well s are stimulated by acidizatio n or hydraulic fracturing .
Hawkins pointed out that if the damaged or stimulated zone is
considered equivalent to an altered zone of uniform permeabili ty ( k s )
and outer radius ( rs ), the additional pressure drop across this zone ( p s )
can be modeled by the steady - state radial flow equation.
29
For non - damaged zone near wellb ore

1   tD 
pD  ln  2   0.80907
2   rD 

For k  k s (damaged zone)
(1) At r  rs or rD 
rs
rw
 






4
k h( pi  prss ) 1   2.64 10 k s t 1 
 s
 ln 
 2   0.80907
2

141.2qB
2 
ctrw
 rs  
 2 
 


r
 

 w 
 pi  prss
 1

 2.64 10  4 k s t 




ln

ln

0
.
80907
  2
      (1)


ct


  rs 

r
( wellbore ) rD  w  1
rw
70.6qB

ks h
(2) At r  rw

k s h( pi  p ws ) 1   2.64  10 4 k s t 


 ln 

0
.
80907


141.2qB
2  
ctrw2



 2.64  10 4 k s t 
70.6qB   1 




 pi  p ws 
ln

ln

0
.
80907
  2
      (2)


k s h   rw 
ct



30
70.6qB
pi  pws 
ks h
 1

 2.64 10 4 k s t 
  0.80907      (2)
ln  2   ln 
ct


  rw 

70.6qB

ks h
 1

 2.64  10  4 k s t 




ln

ln

0
.
80907
  2
      (1)


ct


  rs 

pi  prss
(2)  (1)
2



rs  141.2qB  rs 
70.6qB
ln    
 prss  pws 
ln        (3)
k s h   rw  
ks h
 rw 


Similarity , for k  k (no damage zone)
2



rs  141.2qB  rs 
70.6qB
ln    
 prs  pw 
ln        (4)
k s h   rw  
kh
 rw 


31
Additional pressure drop across damage zone [Eq.(3) - Eq.(4)]
For prss  ps , ps  pw  pws
ps   prss  pws    prs  pw 
  rs 
141 .2qB  k
  1 ln        (1.9)

kh
 ks   rw 
This equation states that the pressure drop in the altered zone is
inversely proportional to ks rather tha n to k .
For k  ks  ps  0
32
With a reduced permeabili ty (damage) or enhanced permeabili ty (stimulate ).
The pressure drop at the wellbore is [from Eq.(1.7d) and Eq.(1.9)]
pi  p wf
70.6qB   1688ct rw2 
  p s

ln 
kh  
kt

 pi  p wf
  rs 
70.6qB   1688ct rw2  141.2qB  k




 1 ln  
ln 



kh  
kt
kh
 ks
  rw 

  rs 
70.6qB   1688ct rw2   k
  2  1 ln  

ln 
kh  
kt
  rw 
  ks
or
pi  p wf

70.6qB   1688ct rw2 



ln

2
s
 
      (1.11)

kh  
kt


k
 r 
s    1 ln  s       (1.10)
 ks
  rw 
If well is damaged , k s  k , s  0
no upper lim it for s
where
if well is stimulated , k s  k , s  0
lower lim it  7or  8
33
From Eq.(1.11)
pi  pwf

qB   1688ctrw2 


 70.6
ln

2
s



kh  
kt



qB   1688ctrw2 
2 s
  ln e 
 70.6
ln 
kh  
kt




2
qB  1688ct rwe  s  
 70.6

ln 
kh  
kt
 

qB   1688ctrs2 

 70.6
ln 
kh  
kt

where
rs  rwe  s  effective wellbore radius
1  rD2 

 pD   Ei  
2  4t D 
or
pD 
where t D 
2.64 10  4 kt
ct rwe  s 
2

1   tD 


ln

0
.
80907


2   rD2 

34
Example 1.1  Calculation of pressures beyond the wellbore u sin g
the Ei  function solution
Given : The well is poducting only oil
q  20 STB / D
  0.72 cp
rw  0.5 ft
k  0.10 md
Bo  1.475 RB / STB
ct  1.5  10 5 psi 1
h  150 ft
pi  3000 psi
  0.23
re  3000 ft
s0
Calculate (1) p  ? at r  1 ft and t  3hrs
(2) p  ? at r  10 ft and t  3hrs
(3) p  ? at r  100 ft and t  3hrs
Solution
2.637 10  4 kt
2.637 10  4  0.1 3
tD 

 127
0.72 1.5 10 5  0.23  (0.5) 2
  ct    rw 2
2
1
r
pD (t D , rD )   Ei ( D )  (1.7a)
2
4t D

1  tD
 ln( 2 )  0.80907  (1.7b)
2  rD

35
r
1
(1) At r  1 ft  rD 

2
rw 0.5
 t D 127

 31.75
 2 
4
 rD

1
22
1
1
3
a ) p D (127,2)   Ei (
)   Ei (7.874  10 )   Ei (0.008)
2
4  127
2
2
1
 (4.259) ( from Table 1.1 or P.4)
2
kh( pi  p ) 1

  4.259  2.1295
141.2qB 2
141.2qB
141.2  20  1.475  0.72
pi  p  2.1295
 2.1295
 425.75
kh
0.1  150
 p  pi  425  3000  425  2575 psi
 1  127
1  tD
 4.266
b) p D (127,2)  ln( 2 )  0.80907   ln( 2 )  0.80907  
 2.133
2  rD
2
2
 2


 pi  p  2.133  199.93  426.47
p  3000  426.47  2573 psi
36
(2) At r  10 ft  rD 
r 10

 20
rw 0.5
 t D 127



0
.
3175
 2

2
20
 rD

2
rD
1
p D (t D , rD )   Ei (
)
2
4t D

1  tD
 ln( 2 )  0.80907
2  rD

1
20 2
1
1
p D (127,20)   Ei (
)  [ Ei (0.787)]  (0.322)
2
4  127
2
2
1  127
 1
 ln( 2 )  0.80907  (0.3382)  0.1691
2  20
 2
( from Table 1.1 on p.4)
37
1
p D (127,20)  (0.322)  0.161
2
kh( pi  p )

 0.161
141.2qB
141.2qB
pi  p  0.161
 0.161 199.93  32.18
kh
 p  3000  32.18  2968 psi
1
p D (127,20)  (0.3382)  0.169
2
pi  p  (0.169)  199.93  33.80
p  3000  33.8  3033.8 psi
38
 tD

127
3
 3.175 10 
 2 
2
200
 rD

r 100
(3) At r  100 ft  rD  
 200
rw 0.5
2
1
rD
1
200 2
1
1
pD (t D , rD )   Ei (
)   Ei (
)  [ Ei (78.7)]  (0)  0
2
4t D
2
4 127
2
2
 p  pi
pD (t D , rD ) 

1  tD
ln(
)

0
.
80907


2  rD 2


1
ln( 3.175 10 3 )  0.80907
2
1
 (4.943)  2.47
2


39
Equation of Diffusivit y
 2 p 1 p
  c 
p

2
r r 2.634  10 4 k t
r
 2 p D 1 p D p D


2
rD rD
t D
rD
kh( pi  p)
0.000264kt
r
where p D 
tD 
rD 
2
141.2qB
rw
crw
40
Solutions
(1) Infinite reservoir
(a) Line source solution
2

1
rD
1  tD
pD   Ei (
)  ln( 2 )  0.80907 
2
4t D
2  rD

(b) Finite wellbore solution
pD 
4
2


0
(1  e
u 2t D
)J1 (u )Y0 (urD )  Y1 (u ) J 0 (urD )du
2
2
u 2 J1 (u )  Y1 (u )


41
Finite wellbore solution
pD
VEH
(t D , rD ) 
4

 2 0
(1  e
u 2t D
)J1 (u )Y0 (urD )  Y1 (u ) J 0 (urD )du
2
2
u 2 J1 (u )  Y1 (u )


At wellbore, rD  1
pD
VEH
(t D ) 
4

 2 0
(1  e )du
2
2
u 3 J1 (u )  Y1 (u )

u 2t D

42
43
Constant rate, Bounded Circular reservoir case
 2 p 1 p
  c 
p


r 2 r r 2.634 10  4 k t
 2 pD 1 pD pD


rD2
rD rD t D
Initial and boundary conditions :
(1) p  pi at t  0 for all r
qB
 p 
( 2)  r 

for t  0

r
2


0
.
001127
kh

 r  rw
 p 
(3)    0 for r  re
 r  r  re
Dimensionl ess form
(1) pD  0 at t D  0 for all rD
 pD 

(2)  rD
 1

r
D  rD 1

(3)
r
pD
 0 at rD  e
rD
rw
or
pD
 1
rD
44
Bounded Cylindrica l Reservoir
From diffusivit y equation of Eq.(1.1)
 2 p 1 p
  c 
p

     (1.1)rw
2
4
r
r r 2.634 10 k t
Boundary conditions and initial conditions :
(1) A well produces at constant rate, qB, into the wellbore
q[] STB / D at surface conditions
B[] RB / STB the formation volume factor
(2) The well, with well bore radius, rw , is centered in a cylindrica l reservoir of radius re ,
and that ther e is no flow across this outer boundary.
(3) Before production begins, the reservoir is at uniform pressure pi .
45
The most useful form of the desired solution relates flowing pressure, p wf , at the
sandface to time and to reservoir rock and fluid properties .
The solution is


e  n t D J 12  n reD 
qB  2t D
3
 pi  141.2
 2  ln reD   2 2 2
      (1.6)
2
kh  reD
4
n 1  n J 1  n reD   J 1  n  

2
p wf

e  n t D J 12  n reD 
2t D
3
p D  2  ln reD   2 2 2
2
4
reD
n 1  n J 1  n reD   J 1  n 

or
where

reD 
2


re
0.000264kt
tD 
rw
ctrw2
the  n are the roots of
J 1  n reD Y1  n   J 1  n Y1  n reD   0
and J 1 and Y1 are Bessel functions
It is an exact solution t o Eq.(1.1) under the assumption s made in its developmen t.
It is sometimes called the van Everdingen - Hurst constant - terminal rate solution
46
(2) Bounded reservoir
e
J12  n reD 
2t D
3
pD  2  ln reD   2 2 2
2


reD
4

J

r

J
n 1
n
1
n eD
1  n 

For t De

 n2t D

2.64 10  4 kt

 0.245
2
cre
 re2 
or t D  0.245 2 
 rw 
2t
3
pD  2D  ln reD 
reD
4
948cre2
or t 
k
47
Pseudostea ty - state solution
From Eq. (1.6) such as

 2t D

e  n t D J 12  n reD 
3
p D t D , reD     2  ln reD   2 2 2

2
4
 reD
n 1  n J 1  n reD   J 1  n  

2


 rw2 
For t De  t D  2   0.25
 re 
 re2 
or t D  0.25 2   0.25reD2
 rw 
9.48  ct    re2
or t 
k
e
J 12  n reD 
0

2
2
2
n 1  n J 1  n reD   J 1  n 


 n2t D

48
Boundary effect time analyzed from type curves
Closed circular reservoir with reD = 3000 case
11.0
10.5
10.0
9.5
pD
9.0
8.5
8.0
infinite reservoir
reD=3000 (re=1050 ft)
7.5
7.0
The visually deviated point from
type curve analysis
6.5
6.0
1.0E+05
1.0E+06
tD*=1.96*106
1.0E+07
1.0E+08
tD
49
Eq .(1.6) becomes
 2t D
3
p D t D , reD     2  ln reD  
4
 reD
 re
qB  0.000527 kt
or p wf  p i  141.2
 ln 

2
kh    c t    re
 rw

Eq1.12
t
p wf
qB  0.000527 k

 141.2
t
kh    ct    re2
Since Vp  re2 h
or r h 
2
e

 

3
  (1.12)
4 

0.0744qB

2

c



h

r
t
e



Vp

50
p wf
0.0744qB
0.0744qB
0.234qB




t
ct  V p
ct  re2  h  
 Vp 
ct   
 
p wf
0.234qB
or

(1.13)
t
ct  V p

p wf

1


t
Vp


for q, B and ct are constant
This result leads to a form of well testing sometimes, called
reservoir limits testing, which seeks to determine reservoir
size from the rate of pressure decline in a wellbore with ti me.
Another form of Eq (1.12) is useful for some applicatio ns.
It involves replacing original reservoir pressure, p i , with average
pressure, p , with th e drainage volume of the well.
51
Form material balance



V   pi  p ctV p



 t 


qB   5.615   pi  p ctV p


 24 

 t  

5.615qB    pi  p ct   re2  h

 24  
 t 
5
.
615
qB
 

 24 
pi  p 
ct   re2  h




0.0744qBt

 (1.14)
2
  ct  h  re

or pi  p 
0.0744qBt
 (1.14a)
2
  ct  h  re
52
Substituti ng Eq (1.14a) in Eq (1.12)
qB  2t D
3
 2  ln reD    (1.12)
p wf  pi  141.2
kh  reD
4

0.0744qBt
qB  2t D
3

 p wf  p 
 141.2
 ln reD  
2
2

kh  reD
4
  ct  h  re

or p  p wf
qB
 141.2
kh
  re  3 
ln      (1.15)
  rw  4 
To account skin effect
From Eq.(1.12)
p wf
 re  3 
qB  0.000527kt
 pi  141.2
 ln     s  (1.17)

2
kh      ct  re
 rw  4 
From Eq.(1.15)

p  p wf
qB   re  3 
 141.2
ln     s  (1.16)
kh   rw  4 
53
From Eq.(1.17)
p wf
 re
qB  0.000527kt
 pi  141.2
 ln   s

2
kh  ct re
 rw e
 2t D
3
 p D    2  ln reD  
4
 reD
2.64  10 4 kt
where
tD 
2
ct rw e  s


 3
    (1.17 a )
 4 
kh pi  p 
pD 
141.2qB
re
reD 
rw e  s
From Eq.(1.16)
p  p wf
qB   re
 141.2
ln   s
kh   rw e
3
4
kh p  p wf
'
pD 
141.2qB
 3
    (1.16a)
 4 
 p D'  ln reD 
where


reD 
re
rw e  s
54
Further, we can define an average permeabili ty, k J , such that
p  p wf
qB
 141.2
kh
  re  3 
ln     s   (1.16)
  rw  4 
qB
 141.2
kJ h
  re
ln 
  rw
 3
  
 4
Form which,
  re  3 
k ln    
 rw  4 

kJ 
 (1.18)
 re  3
ln     s
 rw  4
For a damaged well (s  0), k J  k true
For s stimulated well (s  0) , k J  k true
55
Sometimes, the permeabili ty of a well can be estimated from
productivi ty - index (PI) measuremen ts.
kJ h
q

J 
p  p wf 141.2 B ln  re   3 

  
4
r

  w
J  Pr oductivity  Index ( PI ), STB / D / psi
This method does not necessaril y provide a good estimate
of formation permeabili ty, k
56
Example 1.2 - Analysis of well form PI test
Given : q  100 STB/D (oil)
p wf (BHP) = 1,500 psi (measured)
p  2,000 psi (pressure survey)
h = 10 ft (log analysis)
re = 1,000 ft
μ = 0.5 cp (at current reservoir pressure )
rw = 0.25 ft
B = 1.5 RB/STB
Estimate : (1) PI  ?
(2) kJ  ?
(3) For k  50md (form core data), does this imply tha t
the well is either damaged or stimulated ? What is
the apparent skin factor ?
57
Solution :
(1) J ( PI ) 
q


p  pwf
( 2)  J 
q

p  pwf

100 STB / D 
 0.2 STB / D / psi
2000 psi  1500 psi
kJ h
  re  3 
141.2 B ln    
  rw  4 
  re  3 
141.2qB ln    
 rw  4 

 kJ 


h p  pwf 


  1000  3 
141.2  100   1.5  0.5ln 
 
  0.25  4   16md

102,000  1,500
58
(3) a) k core (= 50md) ﹥k J (= 16md) damaged (badly)
b) Form Eq(1.18)
  re  3 
k ln    
 rw  4 

kJ 
  re  3 
ln     s 
  rw  4 
 re 3
 k
  ln   s  
 rw 4
 kJ
  re  3 
ln    
  rw  4 
  re  3   re 3 
ln      ln  
  rw  4   rw 4 
k
 re 3 
   1 ln  
 kJ
 rw 4 

 50    1,000 
   1 ln 
  0.75  16
 16    0.25 

k
s
kJ
59
Flow Equation for Generalized Reservoir Geometry
Pseudo - steady state solution for a well centered in a circular
drainage area is

p  p wf
qB   re  3 
 141.2
ln     s   (1.16)
kh   rw  4 
A similar equation models pseudo - steady state flow in more
general reservoir shapes :

p  p wf
qB  1  10.06 A  3 
   s   (1.20)
 141.2
 ln 
2 
kh  2  C A rw  4 
Where A  drainage area, ft 2
C A  shape factor for specific drainage area shape and well
location, dimensionl ess. (Table 1.2)
60

J
q

p  pwf

0.00708kh
 1  10.06 A  3 
   s
B  ln 
2 
 2  C A rw  4 
 (1.21)
61
62
Table 1.2 (p.9)
(a) last column : the max time a reservoir is infinite acting
 rw2 
t DA  t D    x
 A
where
or
t
    ct  A  x
2.64  10 4 k
2.64  10 4 kt
t DA 
    ct  A
for a circular reservoir
(b) next to last column :
Time required for the pseudosteady - state equation to be
accurate within 1%
    ct  A
t
x
4
2.64  10 k
(c) The third column from the last column :
Time required for the pseudosteady - state equation to be exact.
t
  c   A
2.64  10 k
4
x
63
Boundary effect time analyzed from type curves
Closed circular reservoir with reD = 3000 case
11.0
10.5
10.0
9.5
pD
9.0
8.5
8.0
infinite reservoir
reD=3000 (re=1050 ft)
7.5
7.0
The visually deviated point from
type curve analysis
6.5
6.0
1.0E+05
1.0E+06
tD*=1.96*106
1.0E+07
1.0E+08
tD
64
Boundary effect time estimated from radius of
investigation equation
closed circular reservoir with reD = 3000 case
11.0
10.5
10.0
(I)
9.5
pD
9.0
( II )
8.5
( III )
8.0
infinite reservoir
reD=3000 (re=1050 ft)
7.5
7.0
The visually deviated point from
type curve analysis
6.5
6.0
1.0E+05
1.0E+06
1.0E+07
1.0E+08
tD
65
Transient Region
pi  pwf

70.6qB   1688ctrw2 
  2 s     (1.11)

ln 
kh  
kt


 pwf is a linear function of log t
Late - transient Region
No simple equation is available to predict th e
relationsh ip between BHP and time in this region
Pseudostea dy - state Region
 1  10.06 A  3 
   s     (1.20)
 ln 
2 
 2  C A rw  4 
or Eq.(1.15) Eq.(1.12) for special case
p  pwf  141.2
qB
kh
66
Late - Transient Region
This region is small ( or, for practical purposes nonexisten t ) for a well
centered in a circular, sequence, or hexagonal drainage area, as Table 1.2
indicates. For a well off - center in its drainage area, the late - transient
region can spane a significan t time region, as Table 1.2 also indicates.
67
Pseudo - steady state flow equation for generalize d reservoir geometry
p wf  pi  141.2
kh pi  p wf
141.2qB
 re  3 
qB  0.000527 kt
   

ln

2
kh  cre
 rw  4 
  0.000527kt  ln  r
 3
 r   4
 w
e
cre2
 re  3
2t D
p D  2  ln   
 re 
 rw  4
 2 
 rw 
pD 
2t D
3

ln
r

eD
4
reD2
re
kh pi  p 
2.634  10 4 kt
reD 
pD 
tD 
rw
141.2qB
crw2
where
 10.06 A 
r
r
10.06(r ) 10.06 A






r

eD
2
2
2 
r
r
(10.06 )rw
C A rw
 C A rw 
C A rw2
1  10.06 A  3

pD 
 2t D   ln 
2 
10.06 A
2  C A rw  4
reD2 
or
2
e
2
w
2
e
2
w
2
e
C A rw2
1  10.06 A  3

pD 
t D  ln 
2 
5.03 A
2  C A rw  4
1
2
68
Pseudo - steady state flow equation for generalize d reservoir geometry
C A rw2
1  10.06 A  3

pD 
t D  ln 
2 
5.03 A
2  C A rw  4
(1) A  re2
C A  31.60
31.60rw2
1  10.06re2  3

pD 
t  ln 
2 D
2 
2  31.60rw  4
5.03re
 rw2  1  re2  3
p D  2t D  2   ln  2  
 re  2  rw  4
re
2
 rw 
 re  3


p D  2t D    ln   
 re 
 rw  4
69
(2) A  xe  xe
C A  30.8828
30.8828rw2
1  10.06 xe2  3

pD 
t D  ln 
2
2 
2  30.8828rw  4
5.03 xe
 rw2 
xe2  3
1 
p D  6.1397 2 t D  ln  0.32574 2  
2 
rw  4
 xe 
 rw2 
 xe
p D  6.1397t D  2   ln 
 rw
 xe 
 rw2 
 xe


p D  6.1397t D  2   ln 
 rw
 xe 
 1
3
  ln( 0.32574) 
4
 2
 rw2 
 xe


p D  6.1397t D  2   ln 
 rw
 xe 

  1.3108

xe
xe

  0.5608  0.75

70
71
Example 1.3 - flow analysis Generalize d Reservoir Geometry
Given :
A  17.42  10 6 ft 2  400acres 
  1cp
ct  1  10 5 psi 1
  0.2  20%
k  100md
72
Calculate :
(1) The time in hours for which
(a) the reservoir is infinite acting
(b) the pseudostea dy - state is exact; and
(c) the pseudostea dy - state is accurate to within 1%
(2) PI and stabilized production rate with p - p wf  500 psi ,
for each of the well in part 1, if
h  10 ft ,
s  3.0 ,
rw  0.3 ft ,
and B  1.2 RB
STB
(3) For the well centered in one of the quadrants of a square,
wri te equations relating constant flow rate and wellbore pressure
drops at elapsed time of 30, 200, and 400 hrs.
Solution :
t DA
 rw2  2.64  10  4 kt  rw2  2.64  10  4 kt
2.64  10  4  100t
 
 t D   

2 

  ct    A 1  1  10 5  0.2  17.42  10 6
 A    ct    rw  A 
 7.577  10  4 t
 t  1320t DA
73
(2) From Eq (1.21 such as)
q
0.00708kh
J

 1.21
 1  10.6 A  3 
p  p wf
   s
B  ln 
2 
 2  C A rw  4 
0.00708  100  10
7.08
J

 1 10.6  17.42  10 6  3 
 1  2.051  10 9
1.2  1   ln 
   3 1.2  ln 
2
CA
2 
 4 
 2  C A  0.3

7.08
7.08

0.6 ln 2.051  10 9  0.6 ln C A  2.7 15.565  0.6 ln C A




  2.25


74
From Eq1.21 such as
J
q
p  p wf


q  p  p wf J  500 J
75
(3) From the results of (1)
For t  30 hrs, the reservoir is in infinite acting period
For t  200 hrs, the reservoir is in late - transient period
(i.e. the pseudostea dy - state equation is not yet accurate)
For t  400 hrs, the pseudostea dy - state equation is accurate
withi n 1% error
Equations
For t  30hrs
pi  p wf
qB
 70.6
kh
  1688    ct  rw2
ln 
kt
 


  2s   1.11


For t  200hrs
no simple equation can be written
For 400hrs
p  p wf
qB
 141.2
kh
 1  10.06 A  3 
   s   1.20 
 ln 
2 
 2  C A rw  4 
76
Radial flow in infinite reservoir with wellb ore storage
Wellbore storage cause

 variable flow rates (variable sandface flow rate)
Consider a shut - in oil well in a reservoir with uniform and unchanging
pressure. Reservoir pressure will support a column of liquid to some
equilibriu m height in the wellbore.
If we open a valve at the surface and initiate flow, the first oil produced will
be that stored in the wellbore, and the initial flow rate from the formation
to the well will be zero.
With increasing flow time , at constant surface producing rate, the amount of
liquid stored in the wellbore will approach a constant v alue.
77
Development of a mathematical relationship between
sandface (formation) and surface flow rates
(a) Example 1 ( changing liquid level )
Mass blance in the wellbore :
mass rate mass rate  Accumulation
 in
   out
   rate


 
 

d  Vwb 
q sf B  qB 


 t   5.61458 
d 
 24 
t []hrs ;Vwb [] ft 3 ; q sf , q []STB / D ; B []RB / STB
78
d  24Vwb 

  q sf  q B
dt  5.615 
24 d
 Awb h   q sf  q B

5.615 dt
24
dh

Awb
 q sf  q B      (1.22)
5.615
dt
gh
sin ce
p w  pt 
     (1.23)
g c 144

pt  surface pressure
lbm ft
lbm

ft
[

]
ft 3 s 2
ft  s 2
lb f 1 ft 2
lbm
gh
1
[ ]
[ ]
gc
ft  s 2 lbm  ft
ft 2 144in 2
lb f  s 2
gh []
 p w  pt 

or
gh
g c 144
d
 p w  pt   g dh      (1.24)
dt
g c 144 dt
dh 144 g c d
 p w  pt       (1.24a)

dt
g dt
79
Substituti ng Eq.(1.24a) in Eq.(1.22), we have
dh 144 g c d
 pw  pt       (1.24a)

dt
g dt
24
dh
Awb
 qsf  q B      (1.22)
5.615
dt
144 g c d

24
 pw  pt   qsf  q B

Awb 
5.615
 g dt

(24)(144 )  g c 
d
  Awb  pw  pt   qsf  q B      (1.25)

5.615  g 
dt
24  144 Awb g c  d
 pw  pt   qsf  q      (1.25a)


B  5.615  g  dt
24Cs d
 pw  pt   qsf  q

B dt
144 Awb g c
where Cs  a wellbore storage cons tan t 
     (1.26)
5.615  g
24Cs d
 pw  pt       (1.27)
 qsf  q 
B dt
For zero or unchanging surface pressure, pt (a major and not
or
d
dpw
 pw  pt  
necessarity valid assumption), i.e.,
dt
dt
24Cs dpw
 qsf  q 
     (1.28)
B dt
80
Dimensionl ess variables
pD
tD
7.08  10 3 kh pi  pw 
     (1.29)

qi B
2.64  10  4 kt
     (1.30)

  c    rw2
where qi  the surface rate at t  0
dpw dp D dt D
dpw
1 dt D dp D


dp D dt dt D
dp D dt D dt
dt
dpw
2.64  10  4 k dp D
1

7.08  10 3 kh   c    rw2 dt D

qi B
qi B
2.64  10  4 k dp D

dt D
c    rw2
7.08  10 3 kh

0.0373qi B dp D
dpw
 1.31

ct    h  rw2 dt D
dt
81
substituting
Eq.1.31 int o
Eq.1.28, we have
 0.0373qi B dp D 
 

2
 ct    h  rw dt D 
0.894qi C s dp D
 qsf  q 
     (1.32)
2
ct    h  rw dt D
24Cs
qsf  q 
B
 0.894C s  dpD
q
 qsf  q  
2  i
 ct    h  rw  dt D
dp D
 qsf  q  CsD qi
dt D
where CsD
0.894Cs

     (1.33)
2
ct    h  rw
q
dp D 
 qsf  qi   CsD
      (1.34)
dt D 
 qi
82
For the constant - rate production , i.e., q (t )  qi  q , Eq.(1.34) becomes

dp 
q sf  q 1  C SD D 
where
dt
D 

q sf
dp
or
 1  C SD D 1.35
q
dt D
0.894Cs
CsD 
ct hrw2
144 Awb g c
Cs 
5.615  g
This is the inner boundary condition for the problem of constant - rate
flow of a slightly compressib ility liquid with well bore storage.
dp D
Note : for small C sD [ i.e., small C s or small Awb ] or for small
dt D
q sf
1
q
The effect of wellbore storage or sandface rate will be negligible .
83
(b) Example 2 (Wellbore being field up)
Consider a wellbore (fig. 1.5) that contains a single - phase
(liquid or gas) and that is produced at surface rate, q.
Mass balance in the wellbore
mass rate mass rate accumulation 
 in
   out
   rate


 
 

d
Vwb cwb pw 
qsf B  qB 
t
d( )
24
where
qsf , q [] STB / D
B [] RB / STB
Vwb [] the volume of wellbore, bbl
cwb [] the compressib ility of the fluid in the wellbore
pw
(at wellbore conditions) , psi 1
[] psi
t [] hr
84
 qsf  q B  24Vwb cwb
dpw
     (1.36)
dt
24Vwb cwb dpw
qsf  q 
     (1.37)
B
dt
C s  cwbVwb      (1.37 a )
or
Let
24C s dpw
     (1.38)
B
dt
Eq.(1.38)  Eq.(1.28)
 qsf  q 
Note : for the gas in the wellbore, C s  cwbVwb 
qsf  q 
or

qsf
q
where
24C s dpw
B
dt
C s  25.645
where
Vwb
 cons tan t
pw
Awb g c
 g
C s  cwbVwb
 1  C sD
dp D
dt D
C sD 
0.894C s
ct hrw2
85
Analytical solutions in fig.1.6 for
the radial diffusivit y equation w ith the wellbore storage equation [Eq.(1.35) ]
From figure 1.6, values of pD (and thus pw ) can be determined for a well in a
formation with given valu es of t D , CsD and s.
Two purposes of figure 1.6 require special mention at this point :
(a) Presence of unit - slop line
(b) End of wellbore storage distortion
86
End of wellbore storage Distortion (fig. 1.6)
When wellb ore storage has ceased (i.e. when q sf  q ), we would expect
the solution t o the flow equations to be the same as if there had never
been any wellbo re storage i.e., the same as for C sD  0.
87
88
Presence of Unit - Slop Line
At earliest t imes for a given valu e of CSD , and for most value s of s , a " unit - slop line "
(i.e. line with 45 slop) is present on the graph.
This line appears remains as long as all production comes from the wellbore and none
comes from the formation.
From Eq.(1.35) such as
qsf
dpD
 1  CSD
 1.35
q
dt D
for qsf  0
 1  CSD
dpD
0
dt D
tD
or
pD
  dt D   CSD dpD
0
0
dt D  CSD dpD      (1.39)
 t D  CsD pD      (1.40)
 log t D  log CsD  log pD      (1.41)
A graph of log pD v.s. log t D will have a slop of unity
or
CSD pD
 1      (1.42)
tD
89
Note that the solutions for finite C sD and for C sD  0 do become
identical after sufficient elapsed time.
One useful empirical observatio n is that this time (called the " end of
wellbore storage distortion " , ) t wbs , occurs approximat ely one and a
half log cycles after the disappeara nce of the unit - slop line.
Another useful observatio n is that the dimensionl ess time at which
wellbore storage distortion ceases is given by
t D  60  3.5s C sD      (1.43)
90
1.4 Radius of investigation
Radius of investigat ion, ri , is the distance that a pressure transient has moved
into a formation flowing a ratio change in a well.
ri  f (formation rock, fluid properties , time elapsed )
91
Two observatio ns are particular ly important
(1) pat
pat
the wellbore
decreases steadily w ith increasing flow time
the fixed value
decreases steadily w ith increasing flow time
(2)The pressure disturbanc e (or pressure transient ) caused by producing
the well moves further into the reservoir as flow time increases.
For the range of flow time shown, there is always a point beyond which
the drawdown in pressure from the original value is negligible .
92
Now consider a well into which we instantane ously inject a volume
of liquid. This injection introduces a pressure disturbanc e into the
formation; the disturbanc e at radius ri will reach its maximum at time t m
after introducti on of the fluid volume. We seek the relationsh ip between
ri and t m . From the solution t o the diffusivit y equation for an instantane ous
line source in an infinite medium (Carslaw, H. S. and Jaeger, J. C., 1959).
The relationsh ip between ri and t m can be derived as follows :
c1  r 2 / 4t
e
t
c1  cons tan t  f (q) ; t  t D
p  pi 
where
t m , a max imum time at ri , can be found by
differentiating and setting equal to zero :
ri 2 948ct ri 2
 tm 

4
k
dp  c1  r 2 / 4t c1r 2  r 2 / 4t
 2 e

e
0
3
dt
t
4t
 kt
 ri  
 948ct
1
2
1
2

 4  2.637 10 kt 
  
      (1.47)
ct



4
93
1
2
1
2
 kt 
 4  2.637  10 4 kt 
  
      (1.47)
ri  
ct
 948ct 


Note : (1) Eq.(1.47) is for q  constant
(2) We also use Eq.(1.47) to calculate the radius of investigat ion
achieved at any time after any rate change in a well.
94
Equivalent
Radius of investigat ion  Radius of drainage
( Advances by Well Test Analysis by Earlougher , R.C.)
rd  0.029
kt
ct
kt

1189ct 
95
 0.5ri 
   ln t D 
log 
 rw 
Many papers give   0.5
 0.5ri
ln 
 rw

  0.5 ln t D

1
0.5ri
 t D2
rw
2
 ri 
2.64  10 4 kt

 
2
2
r


c



r
w
 w
ri 2
2.64  10 4 kt

2
4rw
  c    rw2
4  2.64  10 4 kt
ri 
 4t D rw2
  c 
2
ri  4t D rw2 
kt
     (1.47)
948  c  
96
The uses of the radius of investigat ion
(1) to help explain th e shape of a pressure buildup or pressure drawdown curve :
Earliest t imes - - ri is in the zone of altered permeabili ty, ks, nearest th e wellbore,
Long times - - ri reaches the vicinity of a reservoir boundary or some
massive reservoir heterogene ity.
(2) to provide a guide for well test design :
To estimate the time required to test to the desired depth in the formation.
(3) to provide a means of estimating the length of time required to achieve
stabilized flow, i.e., the time required for a pressure transient to reach the
boundary of a test reservoir,
948ct re2
ts 
     (1.48)
k
This equation applies for a cylindrica l drainage area of radius .
For other drainage - area shapes, time to stabilize can be different,
as illustrate d in Example 1.3
97
Limitation s of the uses of the radius - of - investigat ion :
(1) It is exactly correct only for a homogeneou s, isotropic, cylindrica l
reservoir. Reservoir heterogene ities will decrease the accuracy of
Eq.(1.47)
(2) Eq.(1.47) is exact only for describing the time maximum pressure
disturbanc e reaches radius ri following an instantane ous burst of
injection into or production from a well.
(3) Exact location of ri becomes less well defined for continuous injection
or production at constant rate following a change in rate.
98
Example 1.4 Calculatio n of radius of investigat ion
Given :
k  100md
  0.2
ct  2 10 5 psi 1
  0.5cp
Find (1) t  ? , to run a flow test on an explorator y well for
sufficient ly long to ensure that the well will drain
a cylinder of more than 1,000 ft radius.
(2) q  ?
99
Solution :
(1)
ri  2  1,000 ft  2,000 ft
From eq1.47  such
1
2


kt

ri  
 948    ct 
 kt  948    ct  ri 2
948    ct  ri 2 948  0.2  0.5  2  10 5  (2000) 2
t 

 75.8hrs
k
100
(2)
q  any flows rate sufficient ly large that pressure change with ti me can be
recorded with sufficient precision to be useful for analysis.
100
1.5 The Principle of Superposition
At this point, the Ei - function solution of the most useful solution
to the flow equation appears to be applicable only for describing
the pressure distributi on
(1) in an infinite reservoir, casued by the production of a single well
in the reservoir,
(2) for a production well at constant rate beginning at time zero.
The applicatio n of the superposit ion can remove some of these
restrictio ns.
The principle of superposit ion
The total pressure drop at any point in a reservoir is the sum
of the pressure drops at that point caused by flow in each of
the wells in the reservoir.
101
Ei - function solution
1  rD2
p D t D , rD    Ei  
2  4t D



q   948ct r 2 

Ei 
p  pi  p  70.6
kt
kh


Log - approximat ion solution
rD2
rD2
 100
 0.0025 or
tD
4t D

1   tD 
p D t D , rD   ln  2   0.80907
2   rD 

q  1688  c    rw2 

ln 
p  pi  p  70.6
kt
kh 

102
InterferenceTest
• Consider three wells, well
A, B, and C that start to
produce at the same time
from infinite reservoir (Fig.
1.8). Application of the
principle of superposition
shows that
p
i
 p wf
 p due

total at well
to well
A
A
 p due
to well
B
 p due
to well C
In terms of Ei functions and log arithmic approximations ,
qB   948ct r 2 
      (1.7)
p  pi  70.6
Ei
kh 
kt

or
qB   948ct r 2 

pi  p  70.6
Ei
kh 
kt

103
p  p 
i
wf total at well
A
2


q A B    948ctrw2 
qB B   948ctrAB
  2s A   70.6
 70.6
Ei 
 Ei

kh  
kt
kh
kt




2

qC B   948ctrAC


 70.6
Ei
kh
kt


     (1.49a)
2
2


q A B   1688ctrw A 
qB B   948ctrAB
 70.6
 2s A   70.6
Ei 
ln 


kh  
kt
kh
kt





2

qc B   948ctrAC
 70.6
Ei 

kh
kt


     (1.49)
104
• In Eq.(1.49), there is a skin factor for well A, but does not
include skin factors for wells B and C. Because most wells
have a nonzero skin factor and because we are modeling
pressure inside the zone of altered permeability near well A,
we must include its skin factor.
• However, the pressure of nonzero skin factors for wells B and
C affects pressure only inside their zones of altered
permeability and has no influence on pressure at Well A if
Well A is not within the altered zone of either Well B or Well
C.
105
Bounded reservoir
• Consider the well (in fig. 1.9)
a distance, L, from a single
no-flow boundary.
Mathematically, this problem
is identical to the problem of
a two-well system; actual
well and image well.
p  p 
i
wf total at well
 ( pi  p) due
to well
A
A
 ( pi  p) due
to well
I
2

qB    948ct  rw2 
qB   948  ct 2 L  
  2s A   70.6
 70.6
Ei 
 Ei

kh  
kt
kh
kt




2

qB   1688ct  rw2 
qB   948  ct 2 L  
  2s   70.6
 70.6
Ei 
ln 
    (1.50)
kh  
kt
kh
kt




106
• Extensions of the imaging technique also can be used, for
example, to model
(1) pressure distribution for a well between two boundaries
intersecting at 90°;
(2) the pressure behavior of a well between two parallel
boundaries; and
(3) pressure behavior for wells in various locations completely
surrounded by no-flow boundaries in rectangular-shape
reservoirs.
• [ Matthews, C. S., Brons, F., and Hazebroek, P.: “A method for
determination of average pressure in a bounded reservoir,” Trans,
AIME (1954) 201, 182-191
107
Variable flow-rate
Consider t he case (fig. 1.10) in
which a well produces at rate
q1
0  t  t1
q2
t1  t  t 2
q3
t2  t
What is the pressure at the sandface of the well ?
p
i
 p wf
 p due

total
 ptotal
to well 1
 p due
to well
2
 p due
to well 3

q1 B   1688  ct    rw2 


 70.6
ln
  2 s 
kh  
kt




q 2  q1 B   1688  ct    rw2 


 70.6
ln

2
s
 


kh
k t  t1 
 




q3  q 2 B   1688  ct    rw2 
  2 s 
 70.6
ln 


kh
k
t

t
 

2

     (1.51)
108
• Proceeding in a similar way, we can model an actual well with
dozens of rate changes in its history
• we also can model the rate history for a well with a
continuously changing rate (with a sequence of constant-rate
periods at the average rate during the period).
109
Example 1.5 Use of superposition
Given :
A flowing well is completed in a reservoir that have the
following properties :
pi  2500
h  43
ft
psi
B  1.32 RB / STB
  0.44 cp
ct  18  10 6
  0.16
psi 1
k  25 md
What will the pressure drop be in a shut - in well 500ft from
the flowing well(A) when the flowing well has been shut in for 1 day following
a flow period of 5 days at 300 STB/D?
110
Solutions :
 p  p
i
well B
70.6q1B    948ct r 2 


 Ei
kh
kt

 
70.6q2  q1 B    948ct r 2 


 Ei
kh
  k t  t1  
  948ct r 2 
70.6B 
  948ct r 2 

 pi  p  
  q2  q1 Ei
q1 Ei 
kh 
kt


 k t  t1  
 948ct r 2  948  0.06  0.04  18  10 6  500 
sin ce

 12.01
k
25
70.6B 70.6  0.44  1.32

 0.0381
kh
25  43
2

  12.01 
  12.01 
 pi  p  0.0381300 Ei
  0  300Ei

 6  24 
 1 24 

 0.0381 300Ei 0.0834   Ei 0.500
 11.43 1.993  0.560
 16.37 psi
111
1.6 Horner’s Approximation
• In 1951, Horner reported an approximation that can be used in
many cases to avoid the use of superposition in modeling the
production history of a variable-rate well.
• With this approximation, we can replace the sequence of Ei
functions, reflecting rate changes, with a single Ei function that
contains a single producing time and a single producing rate.
• The single rate is the most recent nonzero rate at which the well
was produced; we call this rate qlast for now.
• This single producing time is found by dividing cumulative
production from the well by the most recent rate; we call this
producing time tp, or pseudoproducing time
t p (hours )  24
cumulative producing from well , N p ( STB)
most recent rate, qlast ( STB / D)
     (1.52)
112
Then, to model pressure behavior at any point in a reservoir, we can
use the simple equation
70.6 qlast B   948ct r 2 
pi  p  
Ei
     (1.53)


kh
kt p


Two questions arise logically at this point :
(1) What is the basis for this approximat ion ?
(2) Under wha t conditions is it applicable ?
113
(1) The basis for the approximation is not rigorous, but intuitive,
and is founded on two criteria:
(a) Use the most recent rate, such a rate, maintained for any
significant period
(b) Choose an effective production time such that the
product of the rate and the production time results in the
correct cumulative production. In this way, material
balance will be maintained accurately.
114
• (2) If the most recent rate is maintained sufficiently
long for the radius of investigation achieved at this rate
to reach the drainage radius of the tested well, then
Horner’s approximation is always sufficiently accurate.
• We find that, for a new well that undergoes a series of
rather rapid rate changes, it is usually sufficient to
establish the last constant rate for at least twice as long
as the previous rate.
• When there is any doubt about whether these guidelines
are satisfied, the safe approach is to use superposition
to model the production history of the well.
115
Example 1.6 – Application of Horner’s Approximation
• Given: the Production history was as follows:
Find : (1) t p  ?
(2) Is Horner' s approximat ion adequate for this case?
If not, how should the production history for this
well be simulated?
116
Solutions :
68 ( STB) 24 (hrs )
 22.7 STB
D
72 (hrs ) 1 ( Day )

Np
52  0  46  68
166
t p  24
 24 
 24 
 175.5 hrs
qlast
22.7
22.7
(1) qlast 
(2)
t last
t next tolast

72 (hrs )
 2.76  2
26 (hrs )
Thus, Horner' s approximat ion is probably adequate for the case.
117
118
Reference Books
•
(A) Lee, J.W., Well Testing, Society of petroleum Engineers of AIME, Dallas, Texas,,
1982.
•
(B) Earlougher, R.C., Jr., Advances in Well Test Analysis, Society of Petroleum
Engineers, Richardson, Texas, 1977, Monograph Series, Vol. 5.
•
(1) Carlson, M.R., Practical Reservoir Simulation: Using, Assessing, and Developing
Results, PennWell Publishing Co., Houston,TX, 2003.
(2) FANCHI, J.R., Principles of Applied Reservoir Simulation, Second Edition,
PennWell Publishing Co., Houston,TX, 2001.
(3) Ertekin, T., Basic Applied Reservoir Simulation, PennWell Publishing Co.,
Houston,TX, 2003.
•
(4) Koederitz, L.F., Lecture Notes on Applied Reservoir Simulation, World Scientific
Publishing
Company, MD, 2005
119
Introduction
• This course intended to explain how to use well
pressures and flow rates to evaluate the formation
surrounding a tested well , by analytical and
numerical methods.
• Basis to this discussion is an understanding of
(1) the theory of fluid flow in porous media, and
(2) pressure-volume-temperature (PVT) relations for
fluid systems of practical interest.
120
Introduction (cont.)
• One major purpose of well testing is to determine the ability of
a formation to produce fluids.
• Further, it is important to determine the underlying reason for
a well’s productivity.
• A properly designed, executed, and analyzed well test usually
can provide information about
FORMATION PERMEABILITY,
extent of WELLBORE DAMAGE (or STIMULATION),
RESERVOIR PRESSURE, and (perhaps)
RESERVOIR BOUNDARIES and
HETEROGENEITIES.
121
Introduction (cont.)
• The basic test method is to create a pressure drawdown in the
wellbore, this causes formation fluids to enter the wellbore.
• If we measure the flow rate and the pressure in the wellbore
during production or the pressure during a shut-in period
following production, we usually will have sufficient
information to characterize the tested well.
122
Introduction (cont.)
• This course discusses
(1) basic equations that describe the unsteady-state flow of fluids in
porous media,
(2) pressure buildup tests,
(3) pressure drawdown tests,
(4) other flow tests,
(5) type-curve analysis,
(6) gas well tests,
(7) interference and pulse tests, and
(8) drillstem and wireline formation tests
• Basic equations and examples use engineering units (field units)
123
Chapter 1 Fluid Flow in Porous Media
124
1.1 Introduction
(a) Discussion of the differential equations that are used most often
to model unsteady-state flow.
(b) Discussion of some of the most useful solutions to these
equations, with emphases on the exponential-integral solution
describing radial, unsteady-state flow.
(c) Discussion of the radius-of-investigation concept
(d) Discussion of the principle of superposition
Superposition, illustrated in multiwell infinite reservoirs, is used
to simulate simple reservoir boundaries and to simulate variable
rate production histories.
(e) Discussion of “pseudo production time”.
125
1.2 The ideal reservoir model
• Assumptions used
(1) Slightly compressible liquid (small and constant
compressibility)
(2) Radial flow
(3) Isothermal flow
(4) Single phase flow
• Physical laws used
(1) Continuity equations (mass balances)
(2) Flow laws (Darcy’s law)
126
Derivation of continuity equation
127
(A) In Cartesian coordinate system
For an infinite small fixed control volume (dx, dy, dz)
with a velocity vector field of V  u, v, w 
128
From Reynolds transpor t theorem
dBsys 
  dV    v  n dA
dt
t CV
CS
 
For Bsys  msys  mass


dmsys
dt
dBsys

dm
1
dm
dm
 

   dV    v  n dA  0
t CV
CS
In a porous media in which porous space and rock matrix exist in a volume of V
dmsys 
 

  d V     v  n dA  0
dt
t CV
CS
where
V [] Apparent volume
V [] True volume

v [] Apparent velocity  Va
A [] Apparent area
129
In the derivation ,
V [] Apparent volume ( pore volume  matrix volume)
v[] Apparent velocity
A[] Apparent area ( pore area  matrix area )

dmsys
dt
 
CV
 
CV

 




 n dA  0
v



V
d



t CV
CS
 

d V     v  n dA  0
t
CS

d V     i vi Ai out    i vi Ai in  0
t
i
i
   i vi Ai in    i vi Ai out  
i
i
CV

d V       (a)
t
 Rate of   Rate of 

 

out
mass

o
int
mass
  Rate of accumulation 
 

 the system   the system 

 

130
For dxdydz  dV  0 and   cons tan t in dV
 
CV

d V    dV     dV    dxdydz
t
t
t
t
or

  





d
V


dxdydz      (b)

t
t
CV
Eq.(a ) and Eq.(b)
  





v
A


v
A

dxdydz      (c)
 i i i in  i i i out
i
i
t
Base on the figure on page 2, the term   i vi Ai  of can be estimated
Face
Inlet mass flow
x
udydz
y
vdxdz
z
wdxdy
outlet mass flow
 u  


u

dx  dydz

x


 v  

v

dy  dxdz


y


 w 


w

dz  dxdy

z

131
Introduce these term into Eq.(c)

 u  
 v  

udydz   u 
dx  dydz  vdxdz   v 
dy  dxdz
x
y




 w 
  

 wdxdy   w 
dz  dxdy 
dxdydz
z
t


  u   v   w 
  
 


dxdydz

dxdydz

y
z 
t
 x
    u   v   w




0
t
x
y
z
  

   v  0
t
  
where
  , , 
v  u , v, w 
x y z
 
132
(B) In Cylindrical Polar Coordinates
For an infinitesi mal fixed control volume dr, rd , dz 
133
From Reynolds transpor t theorem
dBsys 
  dV    v  n dA
dt
t CV
CS
 
For Bsys  msys
dBsys
dm


1
dm
dm
dmsys 
 

   dV    v  n dA  0
dt
t CV
CS
In porous media

d V    i vi Ai out   i vi Ai in  0

t
i
i
CV






v
A


v
A

 i i i in  i i i out  d V       (a)
i
i
CV
t
 Rate of   Rate of 

  
  Rate of accumulation 
 mass in   mass out 
For rdrddz  0

   






d
V



dr

rd


dz




t

t


CV
134
Base on the figure on the top
Face
Inlet mass flow
r  direction
rd  dz
vr rddz
  direction
dr  dz
v drdz
z  direction
dr  rd
vz rddr
outlet mass flow

vr rddz dr
r
1 
v drdz rd
v drdz 
r 

vz rddr  vz rddr dz
z
vr rddz 
135
Introduce these term into Eq.(a)


vr rddz   vr rddz 

vr rddz dr   v drdz
r

1 

v drdz rd 
  v drdz 
r 




 
 v z rddr   v z rddr  v z rddr dz  
rdrd dz
z

 t
1 

v rddrdz   vz rddrdz    rdrddz
   rvr ddrdz 
r 
z
 r
 t
Dividing rddrdz
 1 
vr r   1  v    vz   0


t
r r
r 
z


   v  0
t
1 
rAr   1   A     Az 
where
 A 
A  Ar , A , Az 
r r
r 
z
 
136
For the equation of continuity for cylindrica l coordinate s, such as
 1 
1 


( vr r ) 
( v )  ( v z )  0
t
r r
r 
z
In one - dimensiona l flow (r - direction)
 1 

( vr r )  0
t
r r
In porous media
 1 

( vr r )  0      ( A.2)
t
r r
where   porosity , dim ensionless
vr  sup erficial velocity
 The volumetri c flow rate per unit cross - section
area in the radial direction
137
Flow laws - - Darcy' s law
k p q x
u x  0.001127 x

 x A
k y p q y
u y  0.001127

 y A
k p
u z  0.001127 z (  0.00694  )
 z
RB
RB
where
u[]
q[]
k[]md
2
D
D  ft
[]cp
p[] psi
x, y, z[] ft
138
In radial flow
k p
bbl
vr  0.001127
[ ]
 r
D  ft 2
bbl 5.61458 ft 3
[ ]
D  ft 2
1bbl
k p
[]  6.328 10 3
 r
k p
vr  6.328 10
 r
3
vr  2.634  10
4
k p
 r
ft
( )
D
ft 1Day
D 24hr
 ft 
       ( A.3)
 hr 
139
Darcy and practical units
140
Sub. Eq.(A.3) into Eq.(A.2)
 1 

( vr r )  0      ( A.2)
t
r r
 1   
 4 k p  

  0


r

2
.
634

10
 
t
r r  
 r 
 1
  k p 

 2.634  10 4  r
t
r
r   r 
or
1   rk p 
1

       ( A.5)

 
4
r r   r  2.634  10 t
If
k  const .   const .   const .
Eq.( A.5) becomes
1 k   p 


r




r  r  r  2.634 10  4 t
1   p 

 p
or
r


     ( A.5a )


4
r r  r  2.634  10 k p t
141
For slightly compressib le liquid
c
1 dV
V dp
(m  V ,V 
m

)
m
d  
1 
d 1

 
m dp
dp


c      2
 cdp 
1

 ddp  1 ddp      ( A.6)
d
p

1
po
o

  cdp  
d
  
 c p  po   ln  
 o 
    o e c  p  po       ( A.7)


 

 o e c ( p  po )
p p


  o ce c ( p  po )      ( A.7 a )
p

142
From Eq.( A.5a )
1   p 

 p
     ( A.5a )
 r  
4
r r  r  2.634 10 k p t
1  


c ( p  po ) p 
c ( p  po ) p 

ce
 r o e

o
r r 
r  2.634 10  4 k 
t 
1
  p  p 

c ( p  po ) p
  o e c ( p  po )
 o e c ( p  po ) 

ce
r  
o
r
r  r  r r
2.634 10  4 k
t






1
  p  p
p

c ( p  po ) p
 o e c ( p  po )
 o e c ( p  po ) c 

ce
r  
o
r
r  r  r
r 2.634 10  4 k
t
Dividing by  o e c ( p  po )

1   p   p 
c
p

 r   c  
r r  r   r 
2.634 10  4 k t
2
2
where
 p 
c   0
 r 
because (1)c is very small (2)

p
 pressure gradient is very small
r
1   p 
c
p
r

     ( A.9)


4
r r  r  2.634 10 k t
This is a diffusivit y equation 143