Chapter 5
The Basic Differential Equation for Radial Flow
in a Porous Medium
§ 5.1 Introduction
To derive and to solve the radial fluid flow in porous
medium 1   k   p 
p

r     c    (5.1)
r r  
r 
t
 2 p 1 p c p
 2 

(5.19)
r r
k t
r
1   p  c p
or
(5.20)
r  
r r  r 
k t
§ 5.2 Derivation of the Basic radial differential equation
Assumptions:
-- The reservoir is homogenous in all rock
properties and isotropic with respect to permeability
-- h=const. and hperf=h
-- Single phase fluid
Why not Cartesian geometry?
C.V.
C.V.
Conservation of mass:
Mass flow rate (in) － mass flow rate (out)
＝ Rate of change of mass in the volume element

2  r  h    dr    2  r  h    dr 
t
t
 ( q )

or
 2  r  h  
 (5.2)
r
t
 q  r  dr  q  r 
Using Darcys law for a radial flow
q
(2rh ) k p

r
 2rhk p 


  2rh

r 
t


1   k p 

  

r
 (5.3)
r r   r 
t


r
1   k p 

 r   

(5.3)
r r   r 
t
Since
c
 1 V
m
and  
V p
V
m
 
    1  1  t
c


m p
 p  t p
p 
 c

t t

1   k p 
p
 r     c    (5.1)
r r   r 
t
§ 5.3 Conditions of Solution
p
Radial flow equation： 1   k p 
 r     c   (5.1)
r r   r 
t
The most common and useful analytical solution is for the
initial condition : p  pi
for all r
boundary conditions : q  const . at r  rw
p  pi at r  

constant terminal rate solution (Chapter 7&8)
Radial flow equation：
1   k p 
p
 r     c   (5.1)
r r   r 
t
The three most common conditions
(1) Transient --- Early time; no boundary effect
(Infinite acting reservoir)
p  g (r , t )
p
 f (r , t )
r
(2) Semi steady state --- The effect of the outer boundary has been felt.
p
 0 at r  re
r
and
p
 const . for all r and t  (5.7)
t
 1 dV
V dp
 cVdp  dV
dp
dV
 cV

 q  (5.8)
dt
dt
c
dp
q

dt
cV
dp
q


 (5.10)
dt
c    re2  h  
or
where p  p  average


p   p i Vi

pressure
Vi
vol. avg.
 qi
rate avg.

p   p i qi
(3)Steady state
p  pe  const. at r  re
due to natural water influx or the injection of some fluid and
p
 0 for all r and t
t
§ 5.4 The Linearization of Equation 5.1 for Fluids of small
and constant compressibility
1   k p 
p
 r   c (5.1)
r r   r 
t
1   k 
p k p p k p k  2 p 
p




r

r


r



c




r  r   
r  r r  r  r 2 
t
From Eq.(5.4)
1 
p 
c
 cp    c

 p
r r
1   k 
p k
p p k p k  2 p 
p
      r
 c r


r 2   c 
r  r   
r 
r r  r  r 
t
Note:
 k
p
p
   0 sin ce k ,   f (r ) ;
 0 sin ce
 small
r   
r
r
 2 p 1 p c p
 2 

(5.19)
r r
k t
r
1   p  c p
or
(5.20)
r  
r r  r 
k t
It is for the flow of liquids or for c·p<<1
c in equation(5.20) is the total ,or saturation weighted compressibility
c  ct  c o s o  c w s w  c f
  c  abs  co S o  c w S wc  c f  (5.23)
  abs 1  S wc  
c S
o
o
 c w S wc  c f

(1  S wc )
abs 1  S wc   Effective H .C. porosity
c S
o
o
 c w S wc  c f
1  S wc

 Effective Saturation weighted
k
 diffusivity cons tan t
c
compressib ility
Chapter 6 Well Inflow Equations for
Stabilized Flow Conditions
§ 6.1 Introduction
--- Solutions of the radial diffusivity equation for liquid flow
-- Semi-steady state
p
 const .
t
-- Steady state
p
0
t
§6.2 Semi-Steady state solution
From Eq.(5.8), Such as
cV
p
 q  (5.8)
t
-
cV(Pi - P)  qt (6.1)
p
q

 (5.10)
2
t
c  re h
From Eq(5.10), such as
From Eq(5.20), such as
Eq(5.10)&(5.20)
Integration
1  p
c p
(r ) 
 (5.20)
r r r
k t
1  p
q
(r )  
 (6.2)
2
r r r
  re kh
p
q  r 2
r

 C1 (6.3)
r
2  re2 kh
p
q  r 2
r

 C1 (6.3)
2
r
2  re kh
At
r  re ,
p
0
r

C1 
q
2kh
p
q 1 r


(  2 )  (6.4)
r 2kh r re
Integration
 pr  p wf
Note :
q r  1 r 
Pwf p  2kh rw  r  re2 r
Pr
q  r
r 2 rw2 

ln  2  2  (6.6)
2kh  rw 2re re 
rw2
0
2
re
 pr  p wf
q  r
r 2 rw2 

ln  2  2  (6.6)
2kh  rw 2re re 
at r  re and near well bore with skin
pe  pwf 
PI 

q  re
1
ln(
)


S

 (6.7)
2kh  rw
2

q
2kh

(6.8)
pe  p wf
 re 1

  ln   S 
 rw 2

Since pe & re can not be measured directly,
   pe by p (avg. p within the drainage volume)
re
re
rw
rw
pdV  p 2rhdr

p

 r  r h
dV


re
2
e
2
w
rw

2
p 2
re  rw2
or

p r  p wf
Since

re
rw
2
prdr  2
re

re
rw
q  r
r2
 ln  2

2kh  rw 2re
2 q re  r
r2
 2
r  ln  2

r
re 2kh w  rw 2re

 p  p wf
prdr  (6.10)

  (6.6)



dr  (6.11)

re2 re re2
r
Since  r ln dr 
ln 
rw
rw
2 rw 4
re
re2
r3
& 
dr 
rw 2r 2
8
e
re

 p  p wf

q  re 3


ln   S   (6.12)

2kh  rw 4

$6.3
Steady state solution - - -
1   p 
r   0
r r  r 
§6.4 Example of the Application of the Stabilized
Inflow Equations
Steam injection
Ts
Ts
Temperature
Tr
Tr
rh
rw
rh
Under Steady state
p r  p wf 
q oh
r
ln
2kh
rw
for
rw  r  rh
and
pr  ph 
q oc
r
ln
2kh
rh
for
rh  r  re
q o h 
r 
 ln h   (6.16)
2kh  rw 
q oc  re 
 ln
  (6.17)
at r  re , p e  p h 
2kh  rh 
at r  rh , p h  p wf 
Eq.(6.16)  Eq.(6.17)
q oc   oh
r
r 

ln h  ln e   (6.18)
2kh   oc
rw
rh 
for a stimulated well
p e  p wf 
for an unstimulated
p e  p wf 
well
q oc re
ln  (6.18a)
2kh rw
PI ratio increase 
PI stimulated well

PI unstimulated well  oh
 oc
ln
ln
re
rw
rh
r
 ln e
rw
rh
ln
re
rw
PI stimulated well
PI ratio increase 

PI unstimulated well oh ln rh  ln re
oc rw
rh
u sin g typical field data
Tr  113 F
 oc  980cp
re  382 ft
Ts  525 F
rw  0.23 ft
 oh  3.2cp
rh  65 ft
PI
ratio
382
ln
0.23
increase 
 4.14
3.2
65
382
ln
 ln
980 0.23
65
Exercise 6.1 Wellbore Damage
(1) show the skin factor maybe expressed as
ke  k a ra
ln
ka
rw
S
assume that for
r  ra
steady state
r  ra
semi  steady state
(2) rw  0.33 ft ; re  660 ft
During
drilling
ra  4 ft ; k a  0.01k e (or k e  100k a )
After completion, the well is
stimulated
ra  10 ft ; k a  10k e (or k e  0.1k a )
what
will
be the PI
ratio increase ?
Solution:
(1) steady state
flow
q 
r 
 ln

pr  pwf 
2k a h  rw 
semi  steady state flow
q 
r
r2
 ln
pr  pa 

2k e h  rw 2re2
at
rw  r  ra



ra  r  re
r  ra
q  ra 
 ln
  ( a )
pa  pwf 
2k a h  rw 
at
r  re
pe  pwf
q
 re
1
 ln

   (b)
2k e h  ra
2
Eq.(a )  Eq.(b)
p e  p wf
q
 k e ra
re 1 
 ln  ln  

2k e h  k a rw
ra 2 
re
re k e ra 
q  re 1
 ln   ln  ln  ln 

2k e h  rw 2
ra
rw k a rw 
ke
ra 
q  re 1
 ln   ( - 1)ln 
 p e  p wf 
2k e h  rw 2 k a
rw 
which must be equivalent to
p e  p wf

q  re 1
 ln   S 

2k e h  rw 2

k e  k a ra
S
ln
ka
rw
(2) Before stimulation
k e  k a ra 100k a  k a 
4
S1 
ln

ln
 246.02
ka
rw
ka
0.333
After stimulation
k e  k a ra 0.1k a  k a 
10
S2 
ln

ln
 -3.06
ka
rw
ka
0.333
2k e h
( PI ) A.S
( PI ) B.S
 re 1

r
1

  ln   S 2  ln e   S1 ln 660  1  246.02
rw 2
253.11

  rw 2

 0.333 2

 62.77
2k e h
re 1
660 1
4.0312
ln
  3.06
ln   S 2
0.333 2
rw 2
 r

1
  ln e   S1 
 rw 2

§6.5 Generalized Form of Inflow Equation Under Semi-steady State
Conditions
Semi-steady state equation in terms of the avg. pressure Eq.(6.12)

p  p wf 

q  re 3
 ln   S   (6.12)
2kh  rw 4



3



r
re
q
q 

 ln e  ln e 4  ln e S  

ln
3 
 2kh 
2kh  rw
S

 rw e e 4 

4re2
q  1
q  1
4A


ln
 ln


2kh  2 4 rw e  S 2 e 3 / 2  2kh  2 56.32 rw e  S

q  1
4A

 ln

2

S
2kh  2 r  31.6 rw e

r  1.781 31.6  Dietz Shape factor  C A




where





2

