Pseudo-steady state flow PDF

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Ezeddin Shirif...

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Pseudo-steady State Flow in a Radial System

The physical concept of pseudo-steady state is define as the condition where the pressure at all points in the reservoir changes at the same rate. Mathematically, this condition is given by:

 p (r ,t ) r  constant t

………………………………….(1)

Note: All derivations are in Darcy’s units unless otherwise noted

Physically, this condition is illustrated by

Where Pwfi = wellbore pressure at time, ti Pi = average reservoir pressure at time, ti Pei = external boundary pressure at time, ti Objective: for pseudo-steady state flow conditions 1. Derive a pressure change relation (i.e., ) using the material balance relation. 2. Derive a relation between the average reservoir pressure, P, and the wellbore flowing pressure, pwf. 3. Derive a pressure radius, time (i.e., p(r,t) solution of the radial flow diffusivity equation

Material balance considerations: Recalling the material balance relation for a slightly compressible liquid, we have:

B Np NBi ct

pr  pi 

………………………………….(2)

Or, noting that NBi = Vp we obtain pr  pi 

B Np V Pc t

………………………………….(3)

For a cylindrical reservoir, we have



VP   h re2  rw2



………………………………….(4)

Substituting Eqn. 4 into Eqn. 3 gives us pr  pi 

B N  h r  rw2 ct p



2 e



………………………………….(5)

Recalling the definition of the cumulative production, Np, we have t

N p   q (t )dt

………………………………….(6)

0

Therefore,

dN p q dt

………………………………….(7)

Taking the derivative of Eqn. 5 with respect to time dp r B q  0 2 h re  rw2 c t dt





………………………………….(8)

Note: All derivations are in Darcy’s units unless otherwise noted

Pseudo-steady State Flow Solution for the Radial Flow Diffusivity Equation

The governing partial differential equation for flow in porous media is called diffusivity equation. The diffusivity equation for a slightly compressible liquid is given as 1   p  ct  p r  r r  r  k t

………………………………….(9)

The significant assumption made in Eqn. 9 are:      

Slightly compressible liquid (constant compressibility) Constant fluid viscosity Single phase liquid flow Gravity and capillary pressure are neglected Constant permeability Horizontal radial flow (no vertical flow)

If we assume that the flow rate, q, is constant than

dp dp is also constant, hence is constant dt dt

as well. Assuming q is constant, then

dp dp B   q , 2 dt dt h re  rw2 c t



q  constant



………………………………….(10)

Substituting Eqn. 10 into Eqn. 9 (we note that partial derivatives are now expressed as ordinary), this gives

1   p  ct r  r r  r  k

 B  2 2  h  re  rw ct





 q 

Or reducing to qB 1 d  dp  r  r dr  dr  kh re2  rw2





………………………………….(11)

Defining c

qB kh re 2  rw2





………………………………….(12)

Substituting Eqn.12 into Eqn. 11 we get 1 d  dp  r   c r dr  dr 

………………………………….(13)

Separating

 dp  d r   crdr  dr 

Integrating (indefinite integration)  dp 

 d  r dr   c  rdr Completing

r

dp c   r 2  c1 dr 2

Multiplying through Eqn. 14 by

………………………………….(14)

1 gives us r

dp c c  r 1 dr r 2

………………………………….(15)

For pseudo-steady state we assume a closed reservoir, that is

 dp   dr   0 re Or

c c1  dp   dr   0   2 re  r re e Solving for C1 gives

c c1  re2 2 Substituting Eqn. 16 into Eqn. 15 gives

………………………………….(16)

 dp c  re2   r  2r dr 

………………………………….(17)

Multiplying through Eqn. 17 by dr gives us

dp 

c re2   r dr 2  r 

Integrating across the reservoir, we have r c  re2   dp  2 r  r  r dr pwf w pr

………………………………….(18)

Completing the integration

r c r  re2 ln(r) rw  2 2 

2

p r  p wf

   rw  r

Or p r  pwf 

 r 2  rw2   c 2 r   re ln( )   rw  2   2 

………………………………….(19)

Recalling Eqn. 12 c

qB  kh re2  rw2





………………………………….(12)

Substituting Eqn.12 into Eqn. 19, we obtain

pr  pwf 

 2  1 qB r ln( )  r 2  rw2  2 2  re 2kh re  rw  rw 2 



Expanding through with the

pr  pwf





………………………………….(20)

1 term gives 2 re  rw





2

 

2 2 1 r  rw qB  re2 r  ln( )   2kh  re2  rw2 2 re2  rw2 rw







Eqn. 21 is our final result (in Darcy units)

 

………………………………….(21)

Development of a  p  pwf  Relation for Pseudo-Steady State Flow In this section, we develop the relationship between the average reservoir pressure, p , and the wellbore flowing pressure, p wf ; The definition of the average reservoir pressure is given as

pr

 

r

p rdV

rw

………………………………….(22)

r

 dV rw

And for a cylindrical reservoir, we have





………………………………….(23)

dV  h  (2r )dr

………………………………….(24)

V   h r  rw 2

2

Substituting Eqn. 24 into Eqn. 22 gives pr 

2h h r 2  rw2



rr



rw

pr rdr

Which reduces to pr 

2 r  rw2



2



r

rw

………………………………….(25)

p r rdr

Solving Eqn. 21 for pr gives us

pr  pwf 

 

 

………………………………….(21)

 

 

………………………………….(26)

2 2 1 r  rw qB  re2 r ln( )   2kh  re2  rw2 2 re2  rw2 rw





2 2 qB  re2 r 1 r  rw  pr  pwf  ln( )    rw 2kh  re2  rw2 2 re2  rw2 





Substituting Eqn. 26 into Eqn.25 gives pr 

Separating

2 2 r  rw2





 

2 2  qB  re2 r 1 r  rw ln( )   p rw  wf 2kh  re2  rw2 rw 2 re2  rw2   r





 rdr  

………………………….(27)

2 2 qB  re2 p wf  rdr  2 2 2 2 2 r  rw r  rw 2 kh r e  r w2 rw r

pr 







qB  2 1 2 2 2 r  rw 2 kh 2 re  rw2











r

rw

qB rw2 2 r dr   r2  rw2 2 kh re2  rw2 rw r



3





r

   r ln( r ) dr



w

r

………………………….(28)

  rdr rw

Isolating terms and evaluating each integral, we have r

 rdr  2 r 1

2

 rw2



………………………….(29)

rw r

 r dr  2 r 1

3

4

 rw4



………………………….(30)

rw r

 rw

r r ln( )dr  ? rw

Obviously, the integral of the logarithm term will require a little work to resolve. We could simply look up the appropriate result in a suitable text, but deriving the required result will be enlightning. Starting with the fundamental form of the logarithm integral, we have

x

 x ln( c) dx Integrating by parts

 udv  uv   vdu x u  ln( ) c c du  dx x

dv  xdx v

1 2 x 2

Then

x

1

x

c

 x ln( c ) dx  2 x ln( c )  2  xdx Reducing to

2

1

x

 x ln( c )dx  2 x

2

x c ln( )  x2 c 4

Therefore r



rw r

 rw

r

1  r r rln( ) dr   r 2 ln  rw  rw 2

 1 2   r   4  rw

1  r r rln( ) dr   r 2 ln  rw 2  rw

 1 2   1 2  rw  1 2    r    rw ln   rw   4  2  rw  4 

Or finally, we have

r

 rw

 r  1 1 r rln( ) dr  r 2 ln   r 2  rw2 2 rw  rw  4





………………………….(31)

Substituting Eqn. 29-31 into Eqn. 28 gives

pr 







2 1 qB  2 2 2 r  rw 2 kh 2 re  rw2











   1 r ln  r   1  r  r          2  r  4  2 1 qB r 1  r  r   r  r    2  r  r  2kh r  r  2

qB  re2 2 1 2 2 2 p r r   wf w 2 r 2  rw2 r 2 rw2 2kh re2  rw2 4

2

2

w

2 w

4 w

2

2 w

2 w

2 e

2

2 w

2 w

Reducing to

pr  pwf 

qB re2  1 2  r  1 2 2  r ln    r 2kh r2  rw2 re2  rw2  2  rw  4 qB  2 1 1 2 r  rw2 r 2  rw2 2 2 2 2  2 kh r  rw 2 re  rw  4











 

  r  2 r

2 1 qB  2 2 2 2 kh r  rw 2 re  rw2



 

2 w



2

 rw2

  rw2      







Collecting

pr  pwf  Or finally

qB re2 2kh re2  rw2



 

 

2 2  r2  r  1  qB r  rw qB  rw2      ln   2 2 2 2 2 2    r  rw  rw  2  2kh 4 re  rw 2 kh 2 re  rw









pr  pwf 

  r2 r 2  rw2  r  1 qB  re2 rw2   ln       2  ……………….(32) 2 re2  rw2  2kh  re  rw2  r 2  rw2  rw  2 4 re2  rw2  









 



Eqn. 32 (which is given in Darcy’s units) is our fundamental linking relation between the wellbore and average reservoir pressures during pseudo-steady state flow. However, pr (the average reservoir pressure at a given radius, r) is of little use – except as a rigorous “linking” relation for pressures in the reservoir. In contrast, if we cobsider pr e (i.e., pr at r = re) we obtain the average reservoir pressure based on the entire reservoir volume. Such a result can be directly coupled with the material balance equation to develop a time-pressure relation for pseudo-steady state flow. Evaluating Eqn. 32 at r = re we have p  pre  pwf 

 

  

 re  1  re2  rw2 qB  re2  re2 rw2      ln    2 kh  re2  rw2  re2  rw2  rw  2  4 re2  rw2 2 re2  rw2







  ……….(33) 



Assuming that re>>rw, then

re2 1 re2  rw2





;

r r

 

 rw2 1 2 2 e  rw 2

e



;

rw2  0 correct later on re2  rw2



Substituting these expression into Eqn.33, we obtain

p  pre  pwf 

qB   re  1 1  ln     2kh   rw  2 4 

Or

p  pre  pwf 

qB   re  3 ln    2kh   rw  4 

………………………….(34)

Summarizing our results so far (using generalized units systems) Pressure at any Radius;





r 2  rw2  r qB   re2 pr  pwf  ln      c rkh  re2  rw2  rw  2 re2  rw2 





Average Reservoir Pressure at any Radius;





………………………….(35)

pr  pwf 

r  qB   re 2  r 2 ln     2 2  2 2 c rkh  re  rw  r  rw  rw 







r 2  rw2 rw2 1   2  4 re2  rw2 2 re2  rw2



 



  ……………….(36) 

Average Reservoir Pressure at re (Volumetric Average Pressure);

p  pwf 

qB    re  3  ln     cr kh   rw  4 

………………………….(37)

For a general reservoir geometry, Eqn.37 becomes;

p  pwf 

qB  1  4 A 1    ln cr kh 2  e rw2 CA 

………………………….(38)

Where

 = 0.577216

Euler’s Constant

CA = Dietz shape factor (e.g. CA = 31.62 for circular reservoir

Table of Units Conversion Factors Darcy Units cr

2π

Field Units 3 2 1.127  10  Or 7.081  10 3

SI Units 5 2  8.527  10  Or 5 .358 10  4

Development of a p(r,t) Relation for Pseudo-steady State Flow in Darcy Units Our last objective is to develop a p(r,t) relation for pseudo-steady state flow in a bounded circular reservoir. Recalling the material balance relation (Eqn. 5) we have

pr  pi 

B Np 2 h r   e  rw2 ct





………………………….(42)

For a constant flow rate, q, we have t

N p   q (t )dt

………………………………….(43)

0

Substituting Eqn. 43 into Eqn. 42 p r  pi 

qB t h re 2  rw2 ct





………………………………….(44)

Recalling the average reservoir pressure identity for a well centered in a bounded circular reservoir, we have

p  pwf 

qB    re  3  ln     cr kh   rw  4 

………………………….(45)

Substituting Eqn. 45 into Eqn. 44 gives

pi 

qB qB   re  3  t  p wf  ln    2 2 cr kh   rw  4   h re  rw ct





0r

p wf 

qB   re  3  qB t ln      pi  h  re2  rw2 ct cr kh   rw  4 





Rearranging

pi  pwf  Or

qB    re  3  qB t ln     c rkh   rw  4  h  re2  rw2 ct





………………………….(46)

pi  pwf 

qB    re  3  qB t ln     c rkh   r w  4 V pc t

………………………….(47)

Recalling the wellbore reservoir pressure relation (Eqn. 26), we have (upon slight rearranging)

pi  pwf 

qB   re  3  qB t ln     2kh   r w  4 V pc t

………………………….(47)

 

2 2 1 r  rw qB  re2 r ln( )  pr  pwf   2kh  re2  rw2 2 re2  rw2 rw





 

………………………….(48)

Subtracting Eqn. 48 from Eqn.47 and solving for pr gives us

pr  pi 

 

2 2 qB  re re2 r 3 1 r  rw    ln( ) ln( )  rw 2 re2  rw2 2 kh  rw 4 re2  rw2





  qB t  V c

………………………….(49)

p t

Assuming that re >>rw i.e., ( re2  rw2 )  re2  gives us

 

 

2 2 qB  re 1 r  rw 3 qB pr  pi  t   ln( )  2 kh  rw 2 re2  rw2 4  V p ct

………………………….(50)

Summarizing, we have the following relations in Darcy units

 

 

2 2 3 1 r  rw  qB qB  re re2 r ln( )  pr  pi  t ………………………….(49)  ln( )   2 2 kh  rw 4 re  rw2 rw 2 re2  rw2  V p ct





and

pr  pi 

 

 

2 2 qB  re 1 r  rw 3 qB   t ln( )  2 2 2 kh  rw 2 re  rw 4  V p ct

………………………….(50)

In Field units we have pr  pi  141 . 2

qB  kh

 

 

2 2  re re2 r qB 3 1 r  rw  ln( ) ln( )     t ………….(51)  2 2  5. 615 2 2 rw 2 re  rw  V pc t  rw 4 re  rw





and

 

 

2 2 qB  re qB 1 r  rw 3 5.615 pr  pi 141.2 t ………………………….(52) ln( )  2 2   kh  rw V p ct 2 re  rw 4 

Where time in days and volume in ft3.

For time in hours we use 5.615/24= 0.23395. and

pr  pi  141.2

 

 

2 2 qB  re qB 1 r  rw 3 t ln( ) 0. 23395   2 2    kh  rw V pc t 2 re  rw 4

………………………….(54)

Finally for conditions at the well, we have Darcy Units pwf  pi 

qB   re 3  qB t ln( )     2kh  rw 4  V pc t

………………………….(55)

Field Units (t, days, Vp, ft3) pwf  pi  141.2

qB kh

 re qB 3 t ln( )    5 .615 r V 4 w pc t 

………………………….(56)

Field Units (t, hours, Vp, ft3) pwf  pi  141.2

qB  re 3  qB t ln( )    0.23395  kh  rw 4  V pc t

………………………….(57)

Recall that the pore volume, Vp is given by



VP   h re2  rw2



…………………………….(4)

Illustrations of Pseudo-Steady State Performance in Radial Flow Systems

Figure 1 Radial flow under pseudo-steady state conditions

Figure 2 Radial flow under steady state conditions...