PETE 4052
Lecture 7
Well Testing
Inflow Equations
Spring 2002
March 4
Inflow Equations
Outline
0.
1.
2.
3.
4.
5.
6.
Synopsis
Special Cases of the Diffusivity Equation
Steady-State Inflow in a Circle
Pseudosteady-State Inflow in a Circle
Dietz Shape Factors
Effects of Shape, Skin, and Area
Productivity Index
Related reading
Craft et al., Sections 7.7-7.8, p. 242-248.
Dake: Sections Chapter 6, p. 141-152.
7.0 Synopsis
The flow of a single, compressible fluid through porous, permeable rock can be described using a
partial differential equation known as the diffusivity equation. If the pressure does not change
with time, or if the compressibility is small and constant and the rate is constant, then certain
approximations can be made and we can obtain easy-to-use and very useful inflow equations.
Key Concepts
 Definition of steady state
 Definition of pseudosteady state
 Integration to obtain inflow equations in terms of external pressure and average pressure
 Use of Dietz shape factors
 Productivity Index
7.1 Special Cases of the Diffusivity Equation
In Lecture 6 we derived the diffusivity equation for single-phase flow of an incompressible fluid,
1   p  c t p
............................................... (6.12b)
r  
r r  r 
k t
In this lecture we will consider two particular case:
p
Steady state:
(r , t )  0 .......................................................... (7.1a)
t
and
p
Pseudosteady state:
r, t   constant .................................................... (7.1b)
t
These terms are widely used in reservoir engineering and you should be familiar with them. The
p
third case (in which
varies with r and t) is usually called transient.
t
7.2 Steady-State Inflow in a Circle
Rewriting Eqn. (6.12b) using the steady-state assumption [Eqn. (7.1a)],
1 d  dp 
r
  0 ........................................................ (7.2)
r dr  dr 
We will solve this subject to the boundary conditions
Page 1 of 9
PETE 4052
Lecture 7
Well Testing
Inflow Equations
Spring 2002
March 4
p(rw )  p w .......................................................... (7.3a)
and
p(re )  p e ........................................................... (7.3b)
We will also use the relation, valid for steady-state flow, of
2kh dp
.......................................................... (7.4)
q
r

dr
Note that this is now an ordinary differential equation, because the time derivative has been
eliminated. We can manipulate and then integrate Eqn. (7.2) once to get
 dp 
dr   0
 dr 
dp
r
 C1
dr
Integrating again,
dp C1

dr
r
C1
dp 
dr
r
p  C1 ln( r )  C 2
Applying the boundary conditions at rw and re,
p w  C1 ln( rw )  C 2 or C 2  p w  C1 ln( rw )
r 
p e  C1 ln( re )  C 2  C1 ln  e   p w
 rw 
p  pw
p  pw
C1  e
; C2  pw  e
ln rw 
 re 
 re 
ln  
ln  
 rw 
 rw 
Thus, the equation for pressure is
p(r )  C1 ln( r )  C 2
pe  p w
p  pw
ln( r )  p w  e
ln rw  ................................... (7.5)
 re 
 re 
ln  
ln  
 rw 
 rw 
ln( r / rw )
 p w   pe  p w 
ln( re / rw )
This form makes the linearity of p in ln(r) for steady-state radial flow very clear. The radial
gradient of p is
ln( r / rw )   pe  pw  1
 
 p w   pe  p w 

r 
ln( re / rw )  ln( re / rw ) r
Using Darcy’s Law [Eqn. (7.4)],
2khr dp 2khr  pe  p w  1
q

 dr
 ln( re / rw ) r
......................................... (7.6)
2kh  pe  p w 

 ln( re / rw )

Page 2 of 9
PETE 4052
Lecture 7
Well Testing
Inflow Equations
Spring 2002
March 4
which is just our familiar radial flow formula for incompressible fluids. Note that q is not a
function of r.
We can also express this in terms of the average pressure. For a continuous variable,
re
p
r
pdV
w
re
r
dV
w
We can use Eqn. (7.5) for p to get
ln( r / rw ) 
r 
r  p w  ( pe  p w ) ln( r / r )  2rhdr
e
w 
p 
r
r 2rhdr
w
w
w
.............................. (7.7)
w
( pe  p w ) r
 ln( r / rw )rdr
ln( re / rw ) r

hre2  rw2 
The integral can be evaluated by parts (similar to Dake,
u  ln( r / rw ), dv  r, du  1/ r, and v  r 2 / 2 ),
 re2  rw2 hpw  2h
e
w
p.
r


( pe  p w )  r 2
r r
1
p  pw 
2

h
ln(
r
/
r
)

dr

w 
r 2 
hre2  rw2 
ln( re / rw )  2

r
r
r2 
( pe  p w )  2
1
2
 pw  2
re  rw2  ln( re / rw )  re ln( re / rw )  rw ln( rw / rw )   2  r
( pe  p w )  2
1
1 2
 2 
 pw  2
re  rw2  ln( re / rw ) re ln( re / rw )  2 re  rw 
144;
here
e
e
w
w


e
w



 r2

1
 p w  ( pe  p w )  2 e 2 

 re  rw  2 ln( re / rw ) 
Finally, we have the average pressure equation in terms of p e and p w :
 r2

1
p  p w  ( pe  p w )  2 e 2 
 ................................. (7.8a)
 re  rw  2 ln( re / rw ) 
For a well in a circular drainage area, re  rw and


( p  pw )
p  pw
1
  pe  e
............... (7.8b)
p  pw  ( pe  pw )1 
 e
2 ln( re / rw )
2
 2 ln( re / rw ) 
We can solve Eqn. (7.8b) for  p e  p w 
p  pw
............................................. (7.9a)
pe  p w 


1
1 

 2 ln( re / rw ) 
and use it with Eqn. (7.6) to get an inflow equation in terms of p (assuming, again, that
re  rw ):
p  pw
2kh p  p w 
2kh
........................... (7.9b)
q

1
1
 ln( re / rw ) 1 

 ln( re / rw )  
2 ln( re / rw )
2

Page 3 of 9
PETE 4052
Lecture 7
Well Testing
Inflow Equations
Spring 2002
March 4
Discussion
This development follows Dake’s for pseudo- or semisteady state, p. 141-144, except for steady
state. The final result [Eqn. (7.9b)] is the same as his steady state inflow equation in terms of p ,
given in Table 6.1.
Note that the average pressure [Eqn. (7.8b)] is not the mean of the external and wellbore
pressures. We can find the radius at which this pressure occurs by defining [using Eqn. (7.5)]
ln( r / rw )
.................................... (7.10a)
p  p( r )  p w   pe  p w 
ln( re / rw )
But p is also given by Eqn. (7.8b) so that
( p  pw )
ln( r / rw )
p  pe  e
 p w   pe  p w 
2 ln( re / rw )
ln( re / rw )
pe  p w   pe  p w 
ln( r / rw ) 
ln( r / rw )  ln( re / rw ) 
ln( re / rw )
1
2
ln( r )  ln( rw )  ln( re )  ln( rw ) 
ln( r )  ln( re ) 
r
1
2
............................ (7.10b)
1
2
1
2
re
e
Because e  1.6 and re  rw , the average pressure occurs at a radius greater than the mean
radius (  re / 2 ). The manipulation is simpler if you recognize that ln( r / rw )  ln( re / rw )  1 / 2 .
The average pressure occurs at r  (re  rw ) / 2 because so much of the volume is at high pressure
… a property of being in a radial rather than a linear geometry. For a linear flow geometry, the
average pressure is the mean pressure (and is equal to the pressure halfway through the domain in
linear space). At what point would you expect the pressure to be equal to the average pressure in
spherical coordinates?
7.3 Pseudosteady-State Inflow in a Circle
Definition of Pseudosteady State
Pseudosteady (or, in Dake’s terms, semisteady) state is defined as a regime in which the rate is
constant and the pressure is changing at the same rate everwhere. The rate of pressure change can
be determined by material balance. For a fluid of small and constant compressibility, and
assuming pure depletion drive,
Production  Expansion
or
Rate  time  - (Pore volume)  (Total Compressibility)  (Pressurechange)
because rate was assumed to be constant. The minus sign is needed because for positvie q, the
pressure change must be negative (and all other terms are positive). This is equivalent to writing
qBt  re2 h ct p .............................................. (7.11a)
Page 4 of 9
PETE 4052
Lecture 7
Well Testing
Inflow Equations
Spring 2002
March 4
Dividing by  t and taking the limit as it goes to zero, we get
p
qB
..................................................... (7.11b)
 2
t
re hct
We use partial derivatives because p is still a function of r as well as t.
Discussion
Note that our expression [Eqn. (7.11b)] differs from Dake’s Eqn. (5.10) by a factor B. We have
converted the volume back to a reservoir basis, which is appropriate (a) for a material balance
calculation and (b) because theh compressibilities would be expressed on a reservoir volume
basis. Dake is a bit careless with his B’s and q’s.
We can rewrite Eqn. (7.11b) in terms of dimensionless pressure and time using
kt
kt
t DA 

ct A ct re2
........................................ (7.12a)

 dtDA
k



t t DA dt
ct re2 t DA
and
2k h
 pi  p  .................................................. (7.12b)
pD 
q B
Combining Eqns. (7.12) and (7.11b),
 p D
k
qB
 qB 

 2


2 
re hct
 2kh  ct re  t DA
..................................... (7.13)
p D
 2
t DA
Dimensionless variables are so cool!
Derivation
The derivation of the pseudosteady inflow equations are given in Dake, p. 141-144. The
derivation is very similar to the steady state derivation given in the previous section. I urge you to
review this derivation. The results we will use further are listed below:

qB   re  1
p (r )  p wf 
Pressure at r:
ln     s  ..................................... (7.14a)
2k h   rw  2 
q in terms of pe:
Average pressure p :
q in terms of p :
2k h pe  p w 
.............................................. (7.14b)
  re  1 
B ln     s 
  rw  2 
qB   re  3 
p  pw 
ln     s  ........................................ (7.14c)
2k h   rw  4 
2k h p  p w 
q
.............................................. (7.14d)
  re  3 
B ln     s 
  rw  4 
q
and
r:
r
re
 0.47 re .................................................... (7.14e)
e3 / 4
Page 5 of 9
PETE 4052
Lecture 7
Well Testing
Inflow Equations
Spring 2002
March 4
Several of these formulae are given in Dake’s excellent Table 6.1 (attached at end of these lecture
notes). Note that we have included skin in these expression, following Dake. You should be able
to modify the steady-state equation to include skin, and understand why this works.
Discussion
If we compare the steady-state and pseudosteady-state inflow expressions in terms of average
pressure or in terms of external boundary pressure, we can see the denominator is smaller for the
pseudosteady-state expressions (see Dake, Table 6.1). That is, the same pressure drop moves
more fluid in the pseudosteady state system. We can understand this physically. In steady-state,
all fluid must move all the way from re to rw. On the other hand, in pseudosteady-state, expansion
supplies fluid at all r < re. Thus, less work has to be done to move the fluid and a smaller pressure
drop is required for a given flow rate.
We can rewrite the equations for pressure in dimensionless terms as
p D (rD )  ln( rD ) ..................................................... (7.15a)
Steady-state:
2
r 
p D (rD )  ln( rD )   D  .............................................. (7.15b)
Pseudosteady-state
 reD 
2k h
r
( p  p wf ) and reD  e . Notice that there is a quadratic term in
where we have used p D 
qB
rw
the pseudosteady-state expression in addition to the logarithmic term. This additional term is
insignificant when rD/reD, or near the well. This implies that the gradients near the well must be
similar. On the other hand, far from the well the gradients are different (Figure 7.1).
Comparison of Pressure Profiles
8
Dimensionless Pressure
7
6
5
4
3
2
1
0
1
10
100
Dimensionless Radius
Steady State
Pseudosteady State
Figure 7.1 – Pressure profiles for steady- and pseudosteady-state flow.
Note the following features of Figure 7.1:
Page 6 of 9
1000
PETE 4052
Lecture 7



Well Testing
Inflow Equations
Spring 2002
March 4
The gradient decreases with r for the pseudosteady case. This is true because the gradient at
the outer boundary is zero – it is depletion drive. Thus, the gradient must go from the
p D
 1 at rD = 1 to 0 at reD (in Figure 7.1, reD = 1000).
rD
The decrease in gradient (in log space) for the pseuosteady state implies that the flow rate
decreases. At reD the flow rate is zero. The flow rate rapidly increases as we move inward
from the outer boundary, because a large volume is filled with expanding fluid for each small
change in r at the outer boundary. On the other hand, near the well, only a small volume of
fluid expansion occurs for comparable changes in r, and the flow rate (and thus the gradient)
is nearly constant.
The rate is of course proportional to the gradient of ln(r). For steady state, it is constant. The
dq dV
flow rate for pseudosteady state is given in Figure 7.2. The change flow rate
. It is

dr
dr
parabolic in r, because the volume contributing to the flow is parabolic in (re  r ) . The
gradient is zero at the outer boundary (Fig. 7.1), so the flow rate is zero here also (Fig. 7.2)
Dimensionless Rate, q(rD)/q(1)
1.2
1
0.8
0.6
0.4
0.2
0
0
200
400
600
800
Dimensionless Radius
Figure 7.2 – Variation of flow rate with radius for pseudosteady state flow
Page 7 of 9
1000
PETE 4052
Lecture 7
Well Testing
Inflow Equations
Spring 2002
March 4
7.4 Dietz Shape Factors
The denominator of the pseudosteady inflow equation [Eqn. (7.14b)] can be rearranged to get
r  3
1  r2  3
ln  e    s  ln  e2    s
2  rw  4
 rw  4

3
1  4re2 
4

ln 

ln(
e
)s
2  4rw2 

3
1  4re2 
4

ln 

ln(
e
)s
2  4rw2 
1  4A 
s
 ln 
2  56.32 rw2 

1  4A 
s
ln 
2  31.6rw2 

1  4A 
s
ln 
2  C A rw2 
................................... (7.16)
where  is the exponential of Euler’s constant, and is equal to 1.781 (Euler’s constant is 0.5772).
Why did we do this? This form (the last expression in Eqn. 7.16) is in the Dietz standard form. In
this form, the inflow equation is written as
2k h
q
.............................................. (7.17)
1  4A  
  s
B  ln 
2 
 2  C A rw  
where A is the drainage area and CA is the Dietz shape factor (see Figure 6.4 in Dake). The
amazing thing about the Dietz shape factors is that they can be used for any shape or area of
reservoir. For the most part, these expressions were derived using the method of images, which
we will discuss later in the course. For now, it is sufficient to appreciate that this work has been
done for us, and the Dietz shape factors, in conjunction with Eqn. (7.17), allow as to compute the
stabilized inflow rate for practically any geometry of interest.
Application
The drainage area can often be computed from material balance [Eqn. (7.11b)],
qB
A
......................................................... (7.18)
p
hct
t
In other cases, the area can be computed from maps or pattern spacing. The shape of the well
drainage area can be estimated from maps, using the additional information that drainage volume
is proportional to well rates [as can be deduced from Eqn. (7.11b)]. We will soon see how to
kh
estimate
and s from well tests. The average pressure can also be estimated from well tests.
B
These are all the parameters we need to estimate the stabilized inflow rate. This relationship is
very useful for understanding depletion behavior or reservoirs.
Page 8 of 9
PETE 4052
Lecture 7
Well Testing
Inflow Equations
Spring 2002
March 4
7.5 Effects of Shape, Skin, and Area
The relative importance of several parameters can be deduced from Eqn (7.17), if we use our
knowledge of skin and the properties of logarithms.
First, the basic effects:
 Flow rate decreases as area increases for a given bottomhole pressure and average pressure.
The change in the denominator is logarithmic; increasing A by a factor of e is like adding 1 to
the skin factor.
 Flow rate increases as the shape factor increases; again, the dependence is logarithmic.
Increasing CA by a factor of e is like decreasing the skin factor by 1.
In terms of relative importance of the effects, it would be rare for well drainage areas within a
given reservoir to vary by factors larger than e2, which is like changing s by 2. Examining Figure
6.4, CA varies from about 0.1 for very skewed drainage areas to over 30 for very symmetric areas;
the corresponding change in logarithms is about 5.5 – which is like changing s by 5.5. Thus, it
appears that the effect of area will generally be smaller than the effect of shape. Furthermore, the
effect of shape can (in extreme cases) be as large as the effect of reasonable well-to-well
differences in skin factor. In reservoirs with high skin, the variability of skin will control the
variability in inflow.
Finally, observing flow rates and the rate of change of average pressure gives us no information
about individual parameters such as CA, s, and k. We can only deduce the constant in front of the
pressure difference  p  p w  . This is known as the productivity index.
7.6 Productivity Index
The pseudosteady inflow equation in Dietz form is
p  pw
2kh
.............................................. (7.17)
q
B 1  4 A 
s
ln 
2  C A rw2 
If we define the productivity index, or PI, as the ratio of the flow rate to the pressure drop, then it
is
2k h
PI 
............................................. (7.19)
1  4A  
  s
B  ln 
2 
 2  C A rw  
This leads to a definition of flow efficiency analogous to the previous steady-state radial case, but
now we have
 4A 

ln 
C A rw2 

FE 
.................................................. (7.20)
 4A 
  2s
ln 
2 
 C A rw 
Equation (7.20) shows that the effect of skin on rate depends upon drainage area and shape. Thus,
one cannot determine any of these parameters from pseudosteady- or steady-state data. We will
use pressure transient tests to isolate these parameters.
We will discuss a succession of pseudosteady states and decline curves in Lecture 7A.
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