We assume that drainage area is circular, as shown in Fig 1. The well is located in the centre.
Reservoir has constant height, and overlying and underlying layers are closed. This will be reasonable representative for inflow to most vertical wells.
Figure 2.1: Radial influx
Most reservoirs are more or less heterogeneous. If heterogeneity is evenly distributed and the extent of these is small when compared to reservoir size, the reservoir will likely behave as homogenous. Thus, we assume homogeneous reservoir properties (constant height, constant permeability, etc.)
(or Steady-state flow)
Steady-state means that nothing changes over time. The volume produced from the well - Qw, will be equal to the volume of fluid passed through the reservoir from outer boundary: Qw = Qe.
Flow through the circle around the well, Q(r), will then be constant, regardless of distance. This can quantify that
Q(r) = q o
B o
(2-1) q o
: production (Sm
3
/s)

B o
: Formation volume factor (m
3
/Sm
3
)
With evenly distributed inflow through the perimeter, we will have radial flow towards the well.
Radial Darcy-equation associates fluid flow to pressure v

Q
2
 r h
  k
 o dp
(2-2) dr
By combining and integrating (2-1) and (2-2), we find pressure of the reservoir p
 p w
 q o
2

B o k
 h o r ln r w
(2-3)
Average pressure is generally defined as p
R

1
V
 v p dV (2-4)
For a cylindrical drainage area, the volume i: V=π r
2 h. Entering this and (2-3) into (2-4), we get p
R


1 r e
2 h r e
 o

 p w
 q o
2
B o
 k
 o h r ln r w

 2
 r h dr (2-5)
By integrating (2-5), we can express steady-state productivity index as:
J s
 p
R
 q o
 p w

 o
B o
2


 ln k r e r w h

1
2


(2-6)
From (2-6) follows that if the permeability, viscosity and oil formation volume factor are independent from pressure and rate, the steady-state productivity index will be constant.
(or pseudo-steady-state flow)
Most reservoirs are finite, so that the pressure drops when the oil is tapped. With "pseudo-steady-

state production", we understand that the drop in pressure over the time will be virtually the same everywhere in the reservoir. This is achieved for most finite reservoirs.
As the pressure falls, the reservoir fluid expands. We can quantify this with compressibility equation: c
 
1
V dV dp
(2-7)
C: compressibility of the reservoir fluid
V: volume of reservoir fluid (or volume of fluid in the reservoir - ?) dp: pressure drop dV: change in fluid volume per pressure drop
By pressure reduction, fluid expands and exceeds the reservoir pore volume. Additional fluid volume is production. We can express the rate as a change in volume over the time:
Q
 dV dt
  c V dp
(2-8) dt
Flow into the well corresponds to the production
Q
 
 q o
B o
(2-9)
Since the reservoir is finite, there will be no flow through the outer boundary
Q
  e

0 (2-10)
In between, the flow increase proportionally with the pore volume that contributes
 dQ
 c d V dp dt


 c 2
 h
 dp dt

 rdr (2-11)
Under stationary conditions, the variables inside parenthesis in (2-11) remain a constant. With integrating (2-11) and using (2-9), (2-10), we find the flow distribution

Q
 q o
B o


1


 r r e


2


(2-12)
Figure 2.2 illustrates steady-state and pseudo-steady-state flux (for production: 1000 m 3 /d).
Steady-state flux is constant, while pseudo-steady-state flux declines against the outer limit.
Figure 2.2: Steady-state and pseudo-steady-state influx
To find the pseudo-steady-state productivity index, we first combine and integrate Darcy's equation (2-2) and influx relationship (2-12). This gives the pressure distribution between the wells and the outer boundary p
 p w
 q o
2

B o k
 h o

 ln r r w

1
2 r
2  r w
2 r e
2

 (2-13)
Figure 2.3 compares the pressure at the desktop (2-3) and pseudo-steady-state influx (2-13). The pressure gradient is almost the same at near wellbore (at the well bore it will be equal, since the influxes are the same). Farther out in the reservoir significant differences are seen.

Figure 2.3: Steady-state pressure profile and pseudo-steady-state influx
By calculating the average pressure (2-4) for pseudo-stationary pressure distribution (2-13), we find the pseudo-stationary productivity index. (The calculation is similar to that for stationary influx (2-5), and is left as a challenge for interested readers.) The result, after omitting insignificant contributions, is
J ps

2

 o
B o

 ln k r r e w h

3
4


(2-14)
Equation (2-14) indicates that if the permeability, viscosity and formation volume factor are independent of pressure and rate, it will pseudo-steady-state productivity index will be constant.
After the start-up, or change in rate, constant influx around the well will soon be established.
Gradually, flow and pressure gradient propagate further out. The equation of continuity expresses such changes over the time. In a radial reservoir

1 r

 r
  

 t
  
0 (2-15)
The equation of continuity (2-15), combined with Darcy's flow equation (2-2),
1 r

 r r
 p
 r

 k


1

 
 p


 p
 t

 k c
 p
 t
(2-16)
For homogenous reservoir and constant production, pressure change in time and space can be estimated by solving (2-16). At constant production, until the outer boundary is reached, the solution becomes
1 p ( r , t )
 p i
 q o
4

 o k
B o h
 x y y dy ; there: x

 
4 o k c r w
2 t
(2-17) p i
- initial pressure t- time
(Pa)
(s)
The exponential integral in (2-17) is approximately by series expansion
2
E
1
  ln
 
 n



1
  n n
 n !
x n
γ = Euler’s konstant; ( γ = 1.781....)
We can then write equation of pressure (2-17) p
 p i
 q o
4

 o k
B o h

 ln
4
 t
D

4
1 t
D

1
4


4
1 t
D
There dimensionless time: t
D,
is defined as: t
D


2

...



  o k t c r
2
(2-18)
(2-19)
Figure 2.4 illustrates how the pressure drop, calculated by equation (2-19), propagates through the reservoir

Figure 2.4: Pressure profile by transient flow
After a short time period, at well bore radius, the higher order terms become negligible. Well bore pressure may be approximated from (2-19) p w
   p i
 q

B o o o
4
 k h ln
4 k t
   o c r w
2
 p q K ln Ct i
 o
(2-20)
For a given well the parameter groups: K “transmissivity” and: C “storetivity can be considered constants relate to other. These relate to other reservoir- and fluid parameters as
K


B o o
4
 k h and: C

4 k
   o c r w
2
The simplification above (2-20) is considered acceptable in the time period
From: t
Dw

  o kt c r w
2

100
To: t
De

  kt c r e
2

0 .
1

The lower time limit: t
Dw
, is associated with the omitting of the higher order terms of (2-19). The upper time limit: t
Dw
, is associated with the assumption of infinite reservoir. When the pressure change is noticeable at the outer boundary, the reservoir will no longer behave as infinite.
After a somewhat longer period of time, pressure will stabilize and then fall steadily over the entire drainage area. The time it takes to reach such pseudo-stationary conditions, is associated to dimensionless time as follows t
De

  kt c r e
2

0 .
3 (2-21)
From (2-20), we can formally express "transient productivity index"
J t


 q o

1 p i p w
K ln Ct


B o o
4
 k h

 ln
4 k
   o c r w
2
 ln t


(2-22)
In other words, within transient area the relationship between production and pressure drop will decline over time. Figure 2.5 below shows how transient productivity index approaches pseudosteady-state flow over time.
Figure 2.5: Transient effect on productivity index

In Figure 2.5 is well radius and distance to the outer boundary assumed as follows: r w
= 0.1m
and r e
= 300 m . Transient productivity index is assumed (or calculated with) by (2-22) and pseudo-steady-state by (2-14).
We have previously derived pseudo-steady-state influx (2-12). Since pseudo-steady-state means that the pressure falls steadily in the reservoir, we can express pressure variation in time and space as follows p w
 
 p
R
 
 q o
2

 o k
B o h

 ln r r w e

3
4

 (2-23)
Previously, we derived the relationship between the reservoir and production time (1-9). This is reproduced below (we define now producing as negative) p
R
 p i

B o q o t (2-24) cV
By combining (2-23) and (2-24), we can describe pressure change after pseudo-steady-state conditions are met p w
 p i
 q o
2

 o k
B o h

 ln r r w e

3
4

2 t
De


 p i
 q o
2

 o k
B o h

 ln r r w e

3
4

2 r w
2 r e
2 t
Dw

 (2-25)
We have a relatively simple solution for the transient response (2-20) and pseudo-steady-state response (2-25). In the between period, the response is influenced by the outer boundary, but still not have reached the pseudo-steady-state state. This transition period is defined by the upper limit for the (2-20) and (2-21):
0 .
1
 t
De

0 .
3 (2-26)
Of course, it is also possible to predict pressure response in transition period 3 . However, this is more complicated than the relatively simple relationships presented above. For most reservoirs transition area will be relatively short.

References
1 Mattews, C.S. and Russell, D.G.
Pressure Buildup and Flow Tests in Wells
SPE Monograph vol.1, Henry L. Doherty series
SPE of AIME, N.Y. 1967
2 Abramowitz, M. & Stegun, I. A. (eds.)
Handbook of Mathematical Functions
Dover Publications, inc., NY, 1972
3 van Everdingen, A.F.:
“The skin effect and its influence on the production capacity of a well”
Petr. Trans. of AIME, vol 198. 1953, p 172